The probability that a machine produces a defective item is 0.01. Each item is checked

For each of the following distributions, find \(\mu=\) \(E(X), E[X(X-1)]\), and \(\sigma^{2}=E[X(X-1)]+E(X)-\mu^{2}\) : (a) \(f(x)=\frac{3 !}{x !(3-x) !}\left(\frac{1}{4}\right)^{x}\left(\frac{3}{4}\right)^{3-x}, \quad x=0,1,2,3\). (b) \(f(x)=\frac{4 !}{x !(4-x) !}\left(\frac{1}{2}\right)^{4}, \quad x=0,1,2,3,4\). Equation Transcription: Text Transcription: mu= E(X),E[X(X-1)] sigma^2=E[X(X-1)]+E(X)-mu^2 f(x)=3!/x!(3-x)!(¼)x^ (¾)^3-x, x=0,1,2,3 f(x)=4!/x!(4-x)!(½)^4, x=0,1,2,3,4

Problem 4E Let ? and ?2 denote the mean and variance of the random variable X. Determine E[(X ? ?)/?] and E{[(X ? ?)/? ]2}.

Problem 6E Place eight chips in a bowl: Three have the number 1 on them, two have the number 2, and three have the number 3. Say each chip has a probability of 1/8 of being drawn at random. Let the random variable X equal the number on the chip that is selected, so that the space of X is S = {1, 2, 3}. Make reasonable probability assignments to each of these three outcomes, and compute the mean ? and the variance ?2 of this probability distribution.

Problem 5E Consider an experiment that consists of selecting a card at random from an ordinary deck of cards. Let the random variable X equal the value of the selected card, where Ace = 1, Jack = 11, Queen = 12, and King = 13. Thus, the space of X is S = {1, 2, 3, . . . , 13}. If the experiment is performed in an unbiased manner, assign probabilities to these 13 outcomes and compute the mean ? of this probability distribution.

To find the variance of a hypergeometric random variable in Example 2.3-4 we used the fact that \(E[X(X-1)]=\frac{N_{1}\left(N_{1}-1\right)(n)(n-1)}{N(N-1)}\) . Prove this result by making the change of variables \(k=x-2\) and noting that \(\left(\begin{array}{l}N \\n\end{array}\right)=\frac{N(N-1)}{n(n-1)}\left(\begin{array}{c}N-2 \\ n-2\end{array}\right)\). Equation Transcription: Text Transcription: E[X(X-1)]=N_1(N_1-1)(n)(n-1)/N(N-1) k=x-2 (_n^N)=N(N-1)/n(n-1)N-2 n-2

Find the mean and variance for the following discrete distributions: (a) \(f(x)=\frac{1}{5}, \quad x=5,10,15,20,25\). (b) \(f(x)=1, \quad x=5\). (c) \(f(x)=\frac{4-x}{6}, \quad x=1,2,3\). Equation Transcription: Text Transcription: f(x)=1/5, x=5,10,15,20,25 f(x)=1, x=5 f(x)=4-x/6, x=1,2,3

Find the mean and variance for the following discrete distributions: (a) f(x) = 1 5 , x = 5, 10, 15, 20, 25. (b) f(x) = 1, x = 5. (c) f(x) = 4 x 6 , x = 1, 2, 3.

For each of the following distributions, find = E(X), E[X(X 1)], and 2 = E[X(X 1)]+E(X)2: (a) f(x) = 3! x!(3 x)! 1 4 x3 4 3x , x = 0, 1, 2, 3. (b) f(x) = 4! x!(4 x)! 1 2 4 , x = 0, 1, 2, 3, 4.

Given E(X + 4) = 10 and E[(X + 4)2] = 116, determine (a) Var(X + 4), (b) = E(X), and (c) 2 = Var(X).

Let and 2 denote the mean and variance of the random variable X. Determine E[(X )/] and E{[(X )/] 2}.

Consider an experiment that consists of selecting a card at random from an ordinary deck of cards. Let the random variable X equal the value of the selected card, where Ace = 1, Jack = 11, Queen = 12, and King = 13. Thus, the space of X is S = {1, 2, 3, ... , 13}. If the experiment is performed in an unbiased manner, assign probabilities to these 13 outcomes and compute the mean of this probability distribution.

Place eight chips in a bowl: Three have the number 1 on them, two have the number 2, and three have the number 3. Say each chip has a probability of 1/8 of being drawn at random. Let the random variable X equal the number on the chip that is selected, so that the space of X is S = {1, 2, 3}. Make reasonable probability assignments to each of these three outcomes, and compute the mean and the variance 2 of this probability distribution.

Let X equal an integer selected at random from the first m positive integers, {1, 2, ... , m}. Find the value of m for which E(X) = Var(X). (See Zerger in the references.)

Let X equal the larger outcome when a pair of fair four-sided dice is rolled. The pmf of X is \(f(x)=\frac{2x-1}{16}\), x = 1, 2, 3, 4. Find the mean, variance, and standard deviation of X.

A warranty is written on a product worth $10,000 so that the buyer is given $8000 if it fails in the first year,$6000 if it fails in the second, $4000 if it fails in the third, $2000 if it fails in the fourth, and zero after that. The probability that the product fails in the first year is 0.1, and the probability that it fails in any subsequent year, provided that it did not fail prior to that year, is 0.1. What is the expected value of the warranty?

To find the variance of a hypergeometric random variable in Example 2.3-4 we used the fact that E[X(X 1)] = N1(N1 1)(n)(n 1) N(N 1) . Prove this result by making the change of variables k = x 2 and noting that  N n  = N(N 1) n(n 1)  N 2 n 2

If the moment-generating function of X is \(M(t)=\frac{2}{5} e^{t}+\frac{1}{5} e^{2 t}+\frac{2}{5} e^{3 t}\). find the mean, variance, and pmf of X.

Let X equal the number of people selected at random that you must ask in order to find someone with the same birthday as yours. Assume that each day of the year is equally likely, and ignore February 29. (a) What is the pmf of X? (b) Give the values of the mean, variance, and standard deviation of X. (c) Find P(X > 400) and P(X < 300).

For each question on a multiple-choice test, there are five possible answers, of which exactly one is correct. If a student selects answers at random, give the probability that the first question answered correctly is question 4.

The probability that a machine produces a defective item is 0.01. Each item is checked as it is produced. Assume that these are independent trials, and compute the probability that at least 100 items must be checked to find one that is defective.

Apples are packaged automatically in 3-pound bags. Suppose that 4% of the time the bag of apples weighs less than 3 pounds. If you select bags randomly and weigh them in order to discover one underweight bag of apples, find the probability that the number of bags that must be selected is (a) At least 20. (b) At most 20. (c) Exactly 20.

Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips. (a) Find the pmf of X. Hint: Draw a tree diagram. (b) Find the moment-generating function of X. (c) Use the mgf to find the values of (i) the mean and (ii) the variance of X. (d) Find the values of (i) P(X 3), (ii) P(X 5), and (iii) P(X = 3).

Let X equal the number of flips of a fair coin that are required to observe headstails on consecutive flips. (a) Find the pmf of X. Hint: Draw a tree diagram. (b) Show that the mgf of X is M(t) = e2t /(et 2)2. (c) Use the mgf to find the values of (i) the mean and (ii) the variance of X. (d) Find the values of (i) P(X 3), (ii) P(X 5), and (iii) P(X = 3).

Let X have a geometric distribution. Show that P(X > k + j | X > k) = P(X > j), where k and j are nonnegative integers. Note: We sometimes say that in this situation there has been loss of memory

Given a random permutation of the integers in the set {1, 2, 3, 4, 5}, let X equal the number of integers that are in their natural position. The moment-generating function of X is M(t) = 44 120 + 45 120 et + 20 120 e2t + 10 120 e3t + 1 120 e5t . (a) Find the mean and variance of X. (b) Find the probability that at least one integer is in its natural position. (c) Draw a graph of the probability histogram of the pmf of X.