## 3.6 Exercises

1. Two events each have probability $$0.2$$ of occurring and are independent. What is the probability that neither occur?

2. Two events each have probability $$0.2$$ and are mutually exclusive. What is the probability that neither occur?

3. A smoke-detector system consists of two parts $$A$$ and $$B$$. If smoke occurs then the item $$A$$ detects it with probability $$0.95$$, the item $$B$$ detects it with probability $$0.98$$ whereas both of them detect it with probability $$0.94$$. What is the probability that the smoke will not be detected?

4. Three football players will attempt to kick a field goal. Let $$A_1, A_2, A_3$$ denote the events that the field goal is made by player $$1,2,3$$, respectively. Assume that $$A_1,A_2,A_3$$ are independent and $$P(A_1)= 0.5$$, $$P(A_2)=0.7$$, $$P(A_3)=0.6$$. Compute the probability that exactly 1 player is successful.

5. In a group of $$16$$ candidates, $$7$$ are chemists and $$9$$ are physicists. In how many ways can one choose a group of $$5$$ candidates with $$2$$ chemists and $$3$$ physicists?

6. There is a theorem of combinatorics that states that the number of permutations of $$n$$ objects in which $$n_1$$ are alike of kind $$1$$, $$n_2$$ are alike of kind $$2$$, ..., and $$n_r$$ are alike of kind $$r$$ (that is, $$n=n_1+n_2+\cdots +n_r)$$ is $\frac{n!}{n_1! \cdot n_2! \cdot \cdots \cdot n_r!}.$ Find the number of different words that can be formed by rearranging the letters in the following words.

1. NORMAL

2. HHTTTT

3. LLEWELLYN

4. KITCHISSIPPI

7. A class consists of 490 engineering and 510 science students. The students are divided according to their marks:

Passed Failed
Eng. $$430$$ $$60$$
Sci. $$410$$ $$100$$

If one person is selected randomly, what is the probability that they failed if they were an engineering student?

8. A company which produces a particular drug has two factories, $$A$$ and $$B$$. $$70\%$$ of the drug are made in factory $$A$$, $$30\%$$ in factory $$B$$. Suppose that $$95\%$$ of the drugs produced by factory $$A$$ meet standards while only $$75\%$$ of those produced by factory $$B$$ meet standards. What is the probability that a random dose meets standards?

9. A medical research team wished to evaluate a proposed screening test for Alzheimer’s disease. The test was given to a random sample of $$450$$ patients with Alzheimer’s disease; in $$436$$ cases the test result was positive. The test was also given to a random sample of $$500$$ patients without the disease; only in $$5$$ cases was the result was positive. It is known that in Canada $$11.3\%$$ of the population aged $$65+$$ have Alzheimer’s disease. Find the probability that a person has the disease given that their test was positive (choose the closest answer).

10. Twelve items are independently sampled from a production line. If the probability that any given item is defective is $$0.1$$, the probability of at most two defectives in the sample is closest to …

1. $$0.39$$

2. $$0.99$$

3. $$0.74$$

4. $$0.89$$

11. A student can solve $$6$$ problems from a list of $$10$$. For an exam $$8$$ questions are selected at random from the list. What is the probability that the student will solve exactly $$5$$ problems?

12. Consider the following system with six components. We say that it is functional if there exists a path of functional components from left to right. The probability of each component functions is shown. Assume that the components function or fail independently. What is the probability that the system operates?

13. Pieces of aluminum are classified according to the finishing of the surface and according to the finishing of edge. The results from 85 samples are summarized as follows:

Edge
Surface excellent good
excellent      $60$      $5$
good         $16$      $4$

Let $$A$$ denote the event that a selected piece has an “excellent” surface, and let $$B$$ denote the event that a selected piece has an “excellent” edge. If samples are elected randomly, determine the following probabilities:

1. $$P(A)$$

2. $$P(B)$$

3. $$P(A^c)$$

4. $$P(A\cap B)$$

5. $$P(A\cup B)$$

6. $$P(A^c\cup B)$$

14. Three events are shown in the Venn diagram below.

Shade the region corresponding to the following events:

1. $$A^c$$ $$(A\cap B)\cup (A\cap B^c)$$

2. $$(A\cap B)\cup C$$

3. $$(B\cup C)^c$$

4. $$(A\cap B)^c \cup C$$

15. If $$P(A)=0.1$$, $$P(B)=0.3$$, $$P(C)=0.3$$, and events $$A,B,C$$ are mutually exclusive, determine the following probabilities:

1. $$P(A\cup B\cup C)$$

2. $$P(A\cap B \cap C)$$

3. $$P(A\cap B)$$

4. $$P((A\cup B)\cap C)$$

5. $$P(A^c\cap B^c\cap C^c)$$

6. $$P[(A\cup B\cup C)^c]$$

16. The probability that an electrical switch, which is kept in dryness, fails during the guarantee period, is $$1$$%. If the switch is humid, the failure probability is $$8$$%. Assume that $$90$$% of switches are kept in dry conditions, whereas remaining $$10$$% are kept in humid conditions.

1. What is the probability that the switch fails during the guarantee period?

2. If the switch failed during the guarantee period, what is the probability that it was kept in humid conditions?

17. The following system operates only if there is a path of functional device from left to the right. The probability that each device functions is as shown. What is the probability that the circuit operates?

Assume independence.

18. An inspector working for a manufacturing company has a $$95$$% chance of correctly identifying defective items and $$2$$% chance of incorrectly classifying a good item as defective. The company has evidence that $$1$$% of the items it produces are nonconforming (defective).

1. What is the probability that an item selected for inspection is classified as defective?

2. If an item selected at random is classified as non defective, what is the probability that it is indeed good?

19. Consider an ordinary 52-card North American playing deck ($$4$$ suits, $$13$$ cards in each suit).

1. How many different $$5-$$card poker hands can be drawn from the deck?

2. How many different $$13-$$card bridge hands can be drawn from the deck?

3. What is the probability of an all-spade $$5-$$card poker hand?

4. What is the probability of a flush ($$5-$$cards from the same suit)?

5. What is the probability that a $$5-$$card poker hand contains exactly $$3$$ Kings and $$2$$ Queens?

6. What is the probability that a $$5-$$card poker hand contains exactly $$2$$ Kings, $$2$$ Queens, and $$1$$ Jack?

20. Students on a boat send messages back to shore by arranging seven coloured flags on a vertical flagpole.

1. If they have $$4$$ orange flags and $$3$$ blue flags, how many messages can they send?

2. If they have $$7$$ flags of different colours, how many messages can they send?

3. If they have $$3$$ purple flags, $$2$$ red flags, and $$4$$ yellow flags, how many messages can they send?

21. The Stanley Cup Finals of hockey or the NBA Finals in basketball continue until either the representative team form the Western Conference or from the Eastern Conference wins $$4$$ games. How many different orders are possible ($$WWEEEE$$ means that the Eastern team won in $$6$$ games) if the series goes

1. $$4$$ games?

2. $$5$$ games?

3. $$6$$ games?

4. $$7$$ games?

22. Consider an ordinary 52-card North American playing deck ($$4$$ suits, $$13$$ cards in each suit), from which cards are drawn at random and without replacement, until $$3$$ spades are drawn.

1. What is the probability that there are $$2$$ spades in the first $$5$$ draws?

2. What is the probability that a spade is drawn on the $$6$$th draw given that there were $$2$$ spades in the first $$5$$ draws?

3. What is the probability that $$6$$ cards need to be drawn in order to obtain $$3$$ spades?

4. All the cards are placed back into the deck, and the deck is shuffled. $$4$$ cards are then drawn from. What is the probability of having drawn a spade, a heart, a diamond, and a club, in that order?

23. A student has $$5$$ blue marbles and $$4$$ white marbles in his left pocket, and $$4$$ blue marbles and $$5$$ white marbles in his right pocket. If they transfer one marble at random from their left pocket to his right pocket, what is the probability of them then drawing a blue marble from their right pocket?

24. An insurance company sells a number of different policies; among these, $$60$$% are for cars, $$40$$% are for homes, and $$20$$% are for both. Let $$A_1,A_2,A_3,A_4$$ represent people with only a car policy, only a home policy, both, or neither, respectively. Let $$B$$ represent the event that a policyholder renews at least one of the car or home policies.

1. Compute $$P(A_1)$$, $$P(A_2)$$, $$P(A_3)$$, and $$P(A_4)$$.

2. Assume $$P(B \mid A_1)=0.6$$, $$P(B \mid A_2)=0.7$$, $$P(B \mid A_3)=0.8$$. Given that a client selected at random has a car or a home policy, what is the probability that they will renew one of these policies?

25. An urn contains four balls numbered $$1$$ through $$4$$. The balls are selected one at a time, without replacement. A match occurs if ball $$m$$ is the $$m$$th ball selected. Let the event $$A_i$$ denote a match on the $$i$$th draw, $$i=1,2,3,4$$.

1. Compute $$P(A_i)$$, $$i=1,2,3,4$$.

2. Compute $$P(A_i\cap A_j)$$, $$i,j=1,2,3,4$$, $$i\neq j$$.

3. Compute $$P(A_i\cap A_j\cap A_k)$$, $$i,j,k=1,2,3,4$$, $$i\neq j, i\neq k, j\neq k$$.

4. What is the probability of at least $$1$$ match?

26. The probability that a company’s workforce has at least one accident in a given month is $$(0.01)k$$, where $$k$$ is the number of days in the month. Assume that the number of accidents is independent from month to month. If the company’s year starts on January 1, what is the probability that the first accident occurs in April?

27. A Pap smear is a screening procedure used to detect cervical cancer. Let $$T^-$$ and $$T^+$$ represent the events that the test is negative and positive, respectively, and let $$C$$ represent the event that the person tested has cancer. The false negative rate for this test when the patient has the cancer is $$16$$%; the false positive test for this test when the patient does not have cancer is $$19$$%. In North America, the rate of incidence for this cancer is roughly 8 out of 100,000 women. Based on these numbers, is a Pap smear an effective procedure? What factors influence your conclusion?

28. Of three different fair dice, one each is given to Elowyn, Llewellyn, and Gwynneth. They each roll it. Let $$E=\{\text{Elowyn rolls a 1 or a 2}\}$$, $$LL=\{\text{Llewellyn rolls a 3 or a 4}\}$$, and $$G=\{\text{Gwynneth rolls a 5 or a 6}\}$$ be events.

1. What are the probabilities of each of $$E$$, $$LL$$, and $$G$$ occurring?

2. What are the probabilities of any two of $$E$$, $$LL$$, and $$G$$ occurring simultaneously?

3. What is the probability of all three of the events occurring simultaneously?

4. What is the probability of at least one of $$E$$, $$LL$$, or $$G$$ occurring?

29. Over the course of two baseball seasons, player $$A$$ obtained $$126$$ hits in $$500$$ at-bats in Season 1, and $$90$$ hits in $$300$$ at-bats in Season 2; player $$B$$, on the other hand, obtained $$75$$ hits in $$300$$ at-bats in Season 1, and $$145$$ hits in $$500$$ at-bats in Season 2. A player’s batting average is the number of hits they obtain divided by the number of at-bats.

1. Which player has the best batting average in Season 1? In Season 2?

2. Which player has the best batting average over the 2-year period?

3. Can you explain what is happening here?

30. A stranger comes to you and shows you what appears to be a normal coin, with two distinct sides: Heads ($$H$$) and Tails ($$T$$). They flip the coin $$4$$ times and record the following sequence of tosses: $$HHHH$$.

1. What is the probability of obtaining this specific sequence of tosses? What assumptions do you make along the way in order to compute the probability? What is the probability that the next toss will be a $$T$$.

2. The stranger offers you a bet: they will toss the coin another time; if the toss is $$T$$, they give you $$100\$$, but if it is $$H$$, you give them $$10\$$. Would you accept the bet (if you are not morally opposed to gambling)?

3. Now the stranger tosses the coin 60 times and records $$60\times H$$ in a row: $$H\cdots H$$. They offer you the same bet. Do you accept it?

4. What if they offered $$1000\$$ instead? $$1,000,000\$$?

31. An experiment consists in selecting a bowl, and then drawing a ball from that bowl. Bowl $$B_1$$ contains two red balls and four white balls; bowl $$B_2$$ contains one red ball and two white balls; and bowl $$B_3$$ contains five red balls and four white balls. The probabilities for selecting the bowls are not uniform: $$P(B_1)=1/3$$, $$P(B_2)=1/6$$, and $$P(B_3)=1/2$$, respectively.

1. What is the probability of drawing a red ball $$P(R)$$?

2. If the experiment is conducted and a red ball is drawn, what is the probability that the ball was drawn from bowl $$B_1$$? $$B_2$$? $$B_3$$?

32. Two companies $$A$$ and $$B$$ consider making an offer for road construction. Company $$A$$ submits a proposal. The probability that $$B$$ submits a proposal is $$1/3$$. If $$B$$ does not submit the proposal, the probability that $$A$$ gets the job is $$3/5$$. If $$B$$ submits the proposal, the probability that $$A$$ gets the job is $$1/3$$. What is the probability that $$A$$ will get the job?

33. In a box of $$50$$ fuses there are $$8$$ defective ones. We choose $$5$$ fuses randomly (without replacement). What is the probability that all $$5$$ fuses are not defective?

34. The sample space of a random experiment is $$\{a,b,c,d,e,f\}$$ and each outcome is equally likely. A random variable is defined as follows

outcome $$a$$ $$b$$ $$c$$ $$d$$ $$e$$ $$f$$
$$X$$ 0 0 1.5 1.5 2 3

Determine the probability mass function of $$X$$. Determine the following probabilities:

1. $$P(X=1.5)$$

2. $$P(0.5<X<2.7)$$

3. $$P(X>3)$$

4.$$P(0\le X<2)$$

1. $$P(X=0\;\mbox{\rm or}\;2)$$
35. Determine the mean and the variance of the random variable defined in the previous question.

36. We say that $$X$$ has uniform distribution on a set of values $$\{X_1,\ldots,X_k\}$$ if $P(X=X_i)=\frac{1}{k}, \qquad i=1,\ldots,k .$ The thickness measurements of a coating process are uniformly distributed with values $$0.15$$, $$0.16$$, $$0.17$$, $$0.18$$, $$0.19$$. Determine the mean and variance of the thickness measurements. Is this result compatible with a uniform distribution?

37. Samples of rejuvenated mitochondria are mutated in $$1\%$$ of cases. Suppose $$15$$ samples are studied and that they can be considered to be independent (from a mutation standpoint). Determine the following probabilities:

1. no samples are mutated;

2. at most one sample is mutated, and

3. more than half the samples are mutated.

Use the following CDF table for the $$\mathcal{B}(n,p)$$, with $$n=15$$ and $$p=0.99$$:

38. Samples of $$20$$ parts from a metal punching process are selected every hour. Typically, $$1\%$$ of the parts require re-work. Let $$X$$ denote the number of parts in the sample that require re-work. A process problem is suspected if $$X$$ exceeds its mean by more than three standard deviations.

1. What is the probability that there is a process problem?

2. If the re-work percentage increases to $$4\%$$, what is the probability that $$X$$ exceeds $$1$$?

3. If the re-work percentage increases to $$4\%$$, what is the probability that $$X$$ exceeds $$1$$ in at least one of the next five sampling hours?

39. In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a particular disease. The probability that the person carries a gene is $$0.1$$.

1. What is the probability that $$4$$ or more people will have to be tested in order to detect $$1$$ person with the gene?

2. How many people are expected to be tested in order to detect $$1$$ person with the gene?

3. How many people are expected to be tested in order to detect $$2$$ people with the gene?

40. The number of failures of a testing instrument from contaminated particles on the product is a Poisson random variable with a mean of $$0.02$$ failure per hour.

1. What is the probability that the instrument does not fail in an $$8-$$hour shift?

2. What is the probability of at least $$1$$ failure in a $$24-$$hour day?

41. Use R to generate a sample from a binomial distribution and from a Poisson distribution (select parameters as you wish). Use R to compute the sample means and sample variances. Compare these values to population means and population variances.

42. A container of 100 light bulbs contains $$5$$ bad bulbs. We draw $$10$$ bulbs without replacement. Find the probability of drawing at least $$1$$ defective bulb.

43. Let $$X$$ be a discrete random variable with range $$\{0,1,2\}$$ and probability mass function (p.m.f.) given by $$f(0)=0.5$$, $$f(1)=0.3$$, and $$f(2)=0.2$$. What are the expected value and variance of $$X$$?

44. A factory employs several thousand workers, of whom $$30\%$$ are not from an English-speaking background. If $$15$$ members of the union executive committee were chosen from the workers at random, evaluate the probability that exactly $$3$$ members of the committee are not from an English-speaking background.

45. Assuming the context of the previous questions, what is the probability that a majority of the committee members do not come from an English-speaking background?

46. In a video game, a player is confronted with a series of opponents and has an $$80\%$$ probability of defeating each one. Success with any opponent (that is, defeating the opponent) is independent of previous encounters. The player continues until defeated. What is the probability that the player encounters at least three opponents?

47. Assuming the context of the previous question, how many encounters is the player expected to have?

48. From past experience it is known that $$3\%$$ of accounts in a large accounting company are in error. The probability that exactly $$5$$ accounts are audited before an account in error is found, is:

49. A receptionist receives on average $$2$$ phone calls per minute. Assume that the number of calls can be modeled using a Poisson random variable. What is the probability that he does not receive a call within a $$3-$$minute interval?

50. Roll a $$4-$$sided die twice, and let $$X$$ equal the larger of the two outcomes if they are different and the common value if they are the same. Find the p.m.f. and the c.d.f. of $$X$$.

51. Compute the mean and the variance of $$X$$ as defined in the previous question, as well as $$\textrm{E}[X(5-X)]$$.

52. A basketball player is successful in $$80\%$$ of her (independent) free throw attempts. Let $$X$$ be the minimum number of attempts in order to succeed $$10$$ times. Find the p.m.f. of $$X$$ and the probability that $$X=12$$.

53. Let $$X$$ be the minimum number of independent trials (each with probability of success $$p$$) that are needed to observe $$r$$ successes. The p.m.f. of $$X$$ is $f(x)=P(X=x)=\binom{x-1}{r-1}p^r(1-p)^{x-1},\quad x=r,r+1, \ldots$ The mean and variance of $$X$$ are $\textrm{E}[X]=\frac{r}{p}\quad\mbox{and}\quad \textrm{Var}[X]=\frac{r(1-p)}{p^2}.$ Compute the mean minimum number of independent free throw attempts required to observe 10 successful free throws if the probability of success at the free thrown line is $$80\%$$. What about the standard deviation of $$X$$?

54. If $$n\geq 20$$ and $$p\leq 0.05$$, it can be shown that the binomial distribution with $$n$$ trials and an independent probability of success $$p$$ can be approximated by a Poisson distribution with parameter $$\lambda =np$$: $\frac{(np)^xe^{-np}}{x!}\approx \binom{n}{x}p^x(1-p)^{n-x}.$ A manufacturer of light bulbs knows that $$2\%$$ of its bulbs are defective. What is the probability that a box of $$100$$ bulbs contains exactly at most $$3$$ defective bulbs? Use the Poisson approximation to estimate the probability.

55. Consider a discrete random variable $$X$$ which has a uniform distribution over the first positive $$m$$ integers, i.e. $f(x)=P(X=x)=\frac{1}{m}, \quad x=1,\ldots, m,$ and $$f(x)=0$$ otherwise. Compute the mean and the variance of $$X$$. For what values of $$m$$ is $$\textrm{E}[X]>\textrm{Var}[X]$$?

56. Assume that arrivals of small aircrafts at an airport can be modeled by a Poisson random variable with an average of $$1$$ aircraft per hour.

1. What is the probability that more than $$3$$ aircrafts arrive within an hour?

2. Consider $$15$$ consecutive and disjoint $$1-$$hour intervals. What is the probability that in none of these intervals we have more than $$3$$ aircraft arrivals?

3. What is the probability that exactly $$3$$ aircrafts arrive within $$2$$ hours?

57. In a group of ten students, each student has a probability of $$0.7$$ of passing the exam. What is the probability that exactly $$7$$ of them will pass an exam?

58. A company’s warranty states that the probability that a new swimming pool requires some repairs within the 1st year is $$20\%$$. What is the probability, that the sixth sold pool is the first one which requires some repairs within the 1st year?

59. Consider the following R output:

> pbinom(16,100,0.25)
[1] 0.02111062
> pbinom(30,100,0.25)
[1] 0.8962128
> pbinom(32,100,0.25)
[1] 0.9554037
> pbinom(15,100,0.25)
[1] 0.01108327
> pbinom(17,100,0.25)
[1] 0.03762626
> pbinom(31,100,0.25)
[1] 0.9306511                       

Let $$X\sim\mathcal{B}(n,p)$$ with $$n=100$$ and $$p=0.25$$. Using the R output above, calculate $$P(16\leq X\leq 31)$$.

60. Consider a random variable $$X$$ with probability density function (p.d.f.) given by $f(x)=\begin{cases}0 & \text{if x\leq -1} \\ 0.75(1-x^2) & \text{if -1\leq x<1} \\ 0 & \text{if x\geq 1}\end{cases}$ What is the expected value and the standard deviation of $$X$$?

61. A random variable $$X$$ has a cumulative distribution function (c.d.f.) $F(x)=\begin{cases}0 & \text{if x\leq 0} \\ x/2 & \text{if 0<x<2} \\ 1 & \text{if x\geq 2}\end{cases}$ What is the mean value of $$X$$?

62. Let $$X$$ be a random variable with p.d.f. $$f(x)=\textstyle{\frac{3}{2}}x^2$$ for $$-1\leq x\leq 1$$, and $$f(x)=0$$ otherwise. Find $$P(X^2\leq 0.25)$$.

63. In the inspection of tin plate produced by a continuous electrolytic process, $$0.2$$ imperfections are spotted per minute, on average. Find the probability of spotting at least $$2$$ imperfections in $$5$$ minutes. Assume that we can model the occurrences of imperfections as a Poisson process.

64. If $$X\sim \mathcal{N}(0,4)$$, find the value of $$P(|X| \ge 2.2)$$ using the normal table.

65. If $$X\sim \mathcal{N}(10,1)$$, what is the value of $$k$$ such that $$P(X\le k)=0.701944$$.

66. The time it takes a supercomputer to perform a task is normally distributed with mean $$10$$ milliseconds and standard deviation $$4$$ milliseconds. What is the probability that it takes more than $$18.2$$ milliseconds to perform the task? (use the normal table or R).

67. Let $$X$$ be a random variable. What is the value of $$b$$ (where $$b$$ is not a function of $$X$$) which minimizes $$\textrm{E}[(X-b)^2]$$?

68. The time to reaction to a visual signal follows a normal distribution with mean $$0.5$$ seconds and standard deviation $$0.035$$ seconds.

1. What is the probability that time to react exceeds $$1$$ second?

2. What is the probability that time to react is between $$0.4$$ and $$0.5$$ seconds?

3. What is the time to reaction that is exceeded with probability of $$0.9$$?

69. Refer to the situation described in the aircraft question above.

1. What is the length of the interval such that the probability of having no arrival within this interval is $$0.1$$?

2. What is the probability that one has to wait at least $$3$$ hours for the arrival of $$3$$ aircrafts?

3. What is the mean and variance of the waiting time for $$3$$ aircrafts?

70. Assume that $$X$$ is normally distributed with mean $$10$$ and standard deviation $$3$$. In each case, find the value $$x$$ such that:

1. $$P(X>x)=0.5$$

2. $$P(X>x)=0.95$$

3. $$P(x<X<10)=0.2$$

4. $$P(-x<X-10<x)=0.95$$

5. $$P(-x<X-10<x)=0.99$$

71. Let $$X\sim\text{Exp}(\lambda)$$ with mean $$10$$. Find $$P(X>30\mid X>10)$$.

72. Consider a random variable $$X$$ with the following probability density function:
$f(x)= \left\{ \begin{array}{ll} 0 & \mbox{if x\le -1}\\ \frac{3}{4}(1-x^2) &\mbox{if -1<x<1}\\ 0 &\mbox{if x\ge 1} \end{array} \right.$
What is the value of $$P(X\le 0.5)$$?

73. A receptionist receives on average $$2$$ phone calls per minute. If the number of calls follows a Poisson process, what is the probability that the waiting time for call will be greater than $$1$$ minute?

74. A company manufactures hockey pucks. It is known that their weight is normally distributed with mean $$1$$ and standard deviation $$0.05$$. The pucks used by the NHL must weigh between $$0.9$$ and $$1.1$$. What is the probability that a randomly chosen puck can be used by NHL?

75. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ for the dice example above. Are $$X$$ and $$Y$$ independent?

76. Find $$\textrm{Var}[X_1]$$, $$\textrm{Var}[X_2]$$, and $$\textrm{Cov}(X_1,X_2)$$ for the chip example above. Are $$X_1$$ and $$X_2$$ independent?

77. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ if $$X$$ and $$Y$$ have joint p.m.f. $f(x,y)=\frac{x+y}{21},\quad x=1,2,3,\quad y=1,2.$

78. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ if $$X$$ and $$Y$$ have joint p.m.f. $f(x,y)=\frac{xy^2}{30},\quad x=1,2,3,\quad y=1,2.$ Are $$X$$ and $$Y$$ independent?

79. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ if $$X$$ and $$Y$$ have joint p.m.f. $f(x,y)=\frac{xy^2}{13},\quad (x,y)=(1,1),(1,2),(2,2)$ Are $$X$$ and $$Y$$ independent?

80. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ if $$X$$ and $$Y$$ have joint p.d.f. $f(x,y)=\frac{3}{2}x^2(1-|y|),\quad -1<x<1,\quad -1<y<1.$ Are $$X$$ and $$Y$$ independent?

81. Find $$\textrm{Var}[X]$$, $$\textrm{Var}[Y]$$, and $$\textrm{Cov}(X,Y)$$ if $$X$$ and $$Y$$ follow $f(x,y)=\frac{1}{2\pi}e^{-\frac{1}{2}(x^2+y^2)},\quad -\infty<x<\infty,\quad -\infty<y<\infty.$

82. Suppose that samples of size $$n=25$$ are selected at random from a normal population with mean $$100$$ and standard deviation $$10$$. What is the probability that sample mean falls in the interval $(\mu_{\overline{X}}-1.8\sigma_{\overline{X}},\mu_{\overline{X}}+1.0\sigma_{\overline{X}})?$

83. The amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean $$\mu=8.2$$ minutes and standard deviation $$\sigma=1.5$$ minutes. Suppose that a random sample of $$n=49$$ customers is taken. Compute the approximate probability that the average waiting time for these customers is:

1. Less than $$10$$ min.

2. Between $$5$$ and $$10$$ min.

3. Less than $$6$$ min.

84. A random sample of size $$n_1=16$$ is selected from a normal population with a mean of $$75$$ and standard deviation of $$8$$. A second random sample of size $$n_2=9$$ is taken independently from another normal population with mean $$70$$ and standard deviation of $$12$$. Let $_1$ and $_2$ be the two sample means. Find

1. The probability that $_1- _2$ exceeds $$4$$.

2. The probability that $$3.5< \overline{X}_1- \overline{X}_2<5.5$$.

85. Using R, illustrate the central limit theorem by generating $$M = 300$$ samples of size $$n = 30$$ from:

1. a normal random variable with mean $$10$$ and variance $$0.75$$;

2. a binomial random variable with $$3$$ trials and probability of success $$0.3$$

Repeat the same procedure for samples of size $$n = 200$$. What do you observe?

86. Suppose that the weight in pounds of a North American adult can be represented by a normal random variable with mean $$150$$ lbs and variance $$900$$ lbs$$^2$$. An elevator containing a sign Maximum $$12$$ people’’ can safely carry $$2000$$ lbs. What is the probability that $$12$$ North American adults will not overload the elevator?

87. Let $$X_1,\cdots,X_{50}$$ be an independent random sample from a Poisson distribution with mean $$1$$. Set $$Y=X_1+\cdots+X_{50}$$. Find an approximation of the probability $$P(48 \leq Y \leq 52)$$.

88. A new type of electronic flash for cameras will last an average of $$5000$$ hours with a standard deviation of $$500$$ hours. A quality control engineer intends to select a random sample of $$100$$ of these flashes and use them until they fail. What is the probability that the mean life time of the sample of $$100$$ flashes will be less than $$4928$$ hours?

89. Assume that random variables $$\{X_1,\ldots,X_8\}$$ follow a normal distribution with mean $$2$$ and variance $$24$$. Independently, assume that random variables $$\{Y_1,\ldots,X_{16}\}$$ follow a normal distribution with mean $$1$$ and variance $$16$$. Let $$\overline{X}$$ and $$\overline{Y}$$ be the corresponding sample means. What is $$P(\overline{X}+\overline{Y}>4)$$?

90. Suppose that $$X_1\sim \mathcal{N}(3, 4)$$ and $$X_2\sim \mathcal{N} (3, 45)$$. Given that $$X_1$$ and $$X_2$$ are independent random variables, what is a good approximation of $$P(X_1 + X_2 > 9.5)$$?

91. Consider a sample $$\{X_1,\ldots, X_{10}\}$$ from a normal population $$X_i\sim\mathcal{N}(4,9)$$. Denote by $$\overline{X}$$ and $$S^2$$ the sample mean and the sample variance, respectively. Find $$c$$ such that $P\left(\frac{\overline{X}-4}{S/\sqrt{10}}\leq c\right)=0.99.$