Exam 3: Discrete Random Variables and Probability Distributions

A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis. a. What is the pmf of the number of granite specimens selected for analysis? b. What is the probability that all specimens of one of the two types of rock are selected for analysis? c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

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Question 1

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a. Possible values of X are 5,6,7,8,9,10. (In order to have less than 5 of the granite, there would have to be more than 10 of the basaltic). $P ( X = 5 ) = h ( 5 ; 15,10,20 ) = \frac { \left( \begin{array} { l } 10 \\5\end{array} \right) \left( \begin{array} { l } 10 \\10\end{array} \right) } { \left( \begin{array} { l } 20 \\15\end{array} \right) } = .0163$ Following the same pattern for the other values, we arrive at the pmf, in table form below. $\begin{array} { l c c c c c c } \hline x & 5 & 6 & 7 & 8 & 9 & 10 \\\hline P ( x ) & .0163 & .1354 & .3483 & .3483 & .1354 & .0163 \\\hline\end{array}$ b. P(all 10 of one kind or the other) = P(X = 5) + P(X = 10) = .0163 + .0163 + .0326
c. E(X) = n . ( M / N) = 15 (10 / 20) = 7.5;
V(X) = (5/19) (7.5) [1 - (10/20)] = .9869; $\sigma _ { x }$ = .9934 $\mu \pm \sigma =$ 7.5 $\pm$ .9934 = (6.5066, 8.4934), so we want
P( X = 7) + P( X = 8) = .3483 + .3483 = .6966

Compute the following binomial probabilities directly from the formula for b(x;n,p). a. b(3; 8, .7) b. b(5; 8, .7) c. $P ( 3 \leq X \leq 5 ) \text { when } n = 8 \text { and } p = .7$ d. $P ( X \geq 1 ) \text { when } n = 10 \text { and } p = .1$

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a. $b ( 3 ; 8,7 ) = \left( \begin{array} { l } 8 \\3\end{array} \right) ( .7 ) ^ { 3 } ( .3 ) ^ { 5 } = ( 56 ) ( .00083349 ) = .0467$ b. $b ( 5 ; 8,7 ) = \left( \begin{array} { l } 8 \\5\end{array} \right) ( .7 ) ^ { 5 } ( .3 ) ^ { 3 } = ( 56 ) ( .00453789 ) = .2541$ c. $P ( 3 \leq X \leq 5 ) = b ( 3 ; 8,6 ) + b ( 4 ; 8,6 ) + b ( 5 ; 8,6 ) = .0467 + .1361 + .2541 = .4369$ d. $P ( X \geq 1 ) = 1 - P ( X = 0 ) = 1 - \left( \begin{array} { l } 10 \\0\end{array} \right) ( .1 ) ^ { 10 } ( .9 ) ^ { 10 } = 1 - ( .9 ) ^ { 10 } = .6513$

An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has pmf x 13.5 15.9 19.1 P(x) .2 .4 .4 a. Compute $E ( X ) , E \left( x ^ { 2 } \right) , \text { and } V ( X )$ b. If the price of a freezer having capacity X cubic feet is 25X - 8.5, what is the expected price paid by the next customer to buy a freezer? c. What is the variance of the price 25X - 8.5 paid by the next customer? d. Suppose that although the rated capacity of a freezer is X, the actual capacity is $h ( X ) = X - .01 X ^ { 2 }$ What is the expected actual capacity of the freezer purchased by the next customer?

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Question 3

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a. $E ( X ) = ( 13.5 ) ( .2 ) + ( 15.9 ) ( .4 ) + ( 19.1 ) ( .4 ) = 16.70$ $E \left( X ^ { 2 } \right) = ( 13.5 ) ^ { 2 } ( .2 ) + ( 15.9 ) ^ { 2 } ( .4 ) + ( 19.1 ) ^ { 2 } ( .4 ) = 283.498$ $V ( X ) = 283.498 - ( 16.70 ) ^ { 2 } = 4.608$ b. $E ( 25 X - 8.5 ) = 25 E ( X ) - 8.5 = ( 25 ) ( 16.70 ) - 8.5 = 409$ c. $V ( 25 X - 8.5 ) = V ( 25 X ) = ( 25 ) ^ { 2 } V ( X ) = ( 625 ) ( 4.608 ) = 2880$ d. $E [ h ( X ) ] = E \left[ X - .01 X ^ { 2 } \right] = E ( X ) - .01 E \left( X ^ { 2 } \right) = 16.7 - 2.835 = 13.865$

The pmf for X = the number of major defects on a randomly selected gas stove of a certain type is x 0 1 2 3 4 P(x) .10 .15 .45 .25 .05 Compute the following: a. E(X) b. V(X) directly from the definition c The standard deviation of X d. V(X) using the shortcut formula

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Question 4

If the sample space $\delta$ is an infinite set, does this necessarily imply that any random variable X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.

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Question 5

Twenty-five percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

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Question 6

A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?

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Question 7

Three automobiles are selected at random, and each is categorized as having a diesel (S) or nondiesel (F) engine (so outcomes are SSS, SSF, etc.). If X = the number of cars among the three with diesel engines, list each outcome in S and its associated X value.

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Question 8

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate $\alpha$ = 10 per hour. Suppose that with probability .5 an arriving vehicle will have no equipment violations. a. What is the probability that exactly ten arrive during the hour and all ten have no violations? b. For any fixed $y \geq 10$ what is the probability that y arrive during the hour, of which ten have no violations? c. What is the probability that ten "no-violation" cars arrive during the next hour? [Hint: Sum the probabilities in part (b) from y = 10 to $\infty ,$ ]

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Question 9

Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible. What is the probability that, of 20 randomly chosen drivers coming to an intersection under these conditions, a. At most 6 will come to a complete stop? b. Exactly 6 will come to a complete stop? c. At least 6 will come to a complete stop? d. How many of the next 20 drivers do you expect to come to a complete stop?

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Question 10

Suppose that in one area in California, 40% of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let X denote the number among the four who have earthquake insurance. a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.3)(.7)(.3)(.3) and associated X value 3. There are 15 other outcomes.] b. What is the most likely value for X? c. What is the probability that at least two of the four selected have earthquake insurance?

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Question 11

An insurance company offers its policyholders a number of different payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows: $F ( x ) = \left\{ \begin{array} { l l } 0 & x < 1 \\30 & 1 \leq x < 3 \\40 & 3 \leq x < 4 \\.45 & 4 \leq x < 6 \\.60 & 6 \leq x < 12 \\1 & 12 \leq x\end{array} \right.$ a. What is the pmf of X? b. Using just the cdf, compute $P ( 3 \leq X \leq 6 ) \text { and } P ( X \geq 4 )$ c. Using just the pmf, compute P(X>6).

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Question 12

An automobile service facility specializing in engine tune-ups knows that 50% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 10% on eight-cylinder automobiles. Let X = the number of cylinders on the next car to be tuned. What is the pmf of X?

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Question 13

The n candidates for a store manager have been ranked 1,2,3,…,n. Let X = the rank of a randomly selected candidate, so that X has pmf $p ( x ) = \left\{ \begin{array} { l l } 1 / n & x = 1,2,3 \ldots , n \\0 & \text { otherwise }\end{array} \right.$ (this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The sum of the first n positive integers is n(n + 1)/2, whereas the sum of their squares is n(n + 1)(2n + 1)/6.]

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Question 14

Suppose the number X of tornadoes observed in Kansas during a 1-year period has a Poisson distribution with $\lambda = 9 .$ a. Compute $P ( X \leq 5 )$ b. Compute $P ( 6 \leq X \leq 9 )$ c. Compute $P ( 10 \leq X )$

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Question 15

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?

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Question 16

The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables. a. T = the total number of pumps in use b. X = the difference between the numbers in use at stations 1 and 2 c. U = the maximum number of pumps in use at either station d. Z = the number of stations having exactly two pumps in use

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Question 17

Assume that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 individuals, what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that a. Between 6 and 9 (inclusive) carry the gene. b. At least 10 carry the gene.

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Question 18

The number of tickets issued by a meter reader for parking-meter violations can be modeled by a Poisson process with a rate parameter of five per hour. a. What is the probability that exactly three tickets are given out during a particular hour? b. What is the probability that at least three tickets are given out during a particular hour? c. How many tickets do you expect to be given during a 45-min period?

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Question 19

Let X = the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.

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Question 20

Compute the following binomial probabilities directly from the formula for b(x;n,p). a. b(3; 8, .7) b. b(5; 8, .7) c. $P ( 3 \leq X \leq 5 ) \text { when } n = 8 \text { and } p = .7$ d. $P ( X \geq 1 ) \text { when } n = 10 \text { and } p = .1$

The pmf for X = the number of major defects on a randomly selected gas stove of a certain type is x 0 1 2 3 4 P(x) .10 .15 .45 .25 .05 Compute the following: a. E(X) b. V(X) directly from the definition c The standard deviation of X d. V(X) using the shortcut formula

If the sample space $\delta$ is an infinite set, does this necessarily imply that any random variable X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.

A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?

Three automobiles are selected at random, and each is categorized as having a diesel (S) or nondiesel (F) engine (so outcomes are SSS, SSF, etc.). If X = the number of cars among the three with diesel engines, list each outcome in S and its associated X value.

An automobile service facility specializing in engine tune-ups knows that 50% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 10% on eight-cylinder automobiles. Let X = the number of cylinders on the next car to be tuned. What is the pmf of X?

Suppose the number X of tornadoes observed in Kansas during a 1-year period has a Poisson distribution with $\lambda = 9 .$ a. Compute $P ( X \leq 5 )$ b. Compute $P ( 6 \leq X \leq 9 )$ c. Compute $P ( 10 \leq X )$

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?

Let X = the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.

Overview and Descriptive Statistics

Probability

Continuous Random Variables and Probability Distributions

Joint Probability Distributions and Random Samples

Point Estimation

Statistical Intervals Based on a Single Sample

Tests of Hypotheses Based on a Single Sample

Inferences Based on Two Samples

The Analysis of Variance

Multifactor Analysis of Variance

Simple Linear Regression and Correlation

Nonlinear and Multiple Regression

Goodness-Of-Fit Tests and Categorical Data Analysis

Distribution-Free Procedures

Quality Control Methods

Filters

Exam 3: Discrete Random Variables and Probability Distributions

The pmf for X = the number of major defects on a randomly selected gas stove of a certain type is x 0 1 2 3 4 P(x) .10 .15 .45 .25 .05 Compute the following: a. E(X) b. V(X) directly from the definition c The standard deviation of X d. V(X) using the shortcut formula

If the sample space δ\deltaδ is an infinite set, does this necessarily imply that any random variable X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.

A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?

Three automobiles are selected at random, and each is categorized as having a diesel (S) or nondiesel (F) engine (so outcomes are SSS, SSF, etc.). If X = the number of cars among the three with diesel engines, list each outcome in S and its associated X value.

An automobile service facility specializing in engine tune-ups knows that 50% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 10% on eight-cylinder automobiles. Let X = the number of cylinders on the next car to be tuned. What is the pmf of X?

Suppose the number X of tornadoes observed in Kansas during a 1-year period has a Poisson distribution with λ=9.\lambda = 9 .λ=9. a. Compute P(X≤5)P ( X \leq 5 )P(X≤5) b. Compute P(6≤X≤9)P ( 6 \leq X \leq 9 )P(6≤X≤9) c. Compute P(10≤X)P ( 10 \leq X )P(10≤X)

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?

Let X = the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.

Overview and Descriptive Statistics

Probability

Continuous Random Variables and Probability Distributions

Joint Probability Distributions and Random Samples

Point Estimation

Statistical Intervals Based on a Single Sample

Tests of Hypotheses Based on a Single Sample

Inferences Based on Two Samples

The Analysis of Variance

Multifactor Analysis of Variance

Simple Linear Regression and Correlation

Nonlinear and Multiple Regression

Goodness-Of-Fit Tests and Categorical Data Analysis

Distribution-Free Procedures

Quality Control Methods

Filters

If the sample space $\delta$ is an infinite set, does this necessarily imply that any random variable X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.

Suppose the number X of tornadoes observed in Kansas during a 1-year period has a Poisson distribution with $\lambda = 9 .$ a. Compute $P ( X \leq 5 )$ b. Compute $P ( 6 \leq X \leq 9 )$ c. Compute $P ( 10 \leq X )$