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5.34. In each case, compute the expectation of the random variable X. (a) The values of X are uniformly distributed on the set {0, 1, 2, ..., N-1}. (See Example 5.28.) (b) The values of X are uniformly distributed on the set {1, 2,..., N}. (c) The values of X are uniformly distributed on the set {1, 3, 7, 11, 19, 23). (d) X is a random variable with a binomial density function; see formula (5.23) in Example 5.29 on page 240.

5.34. In each case, compute the expectation of the random variable X. (a) The values of X are uniformly distributed on the set {0, 1, 2, ..., N-1}. (See Example 5.28.) (b) The values of X are uniformly distributed on the set {1, 2,..., N}. (c) The values of X are uniformly distributed on the set {1, 3, 7, 11, 19, 23). (d) X is a random variable with a binomial density function; see formula (5.23) in Example 5.29 on page 240.

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.3: Binomial Probability
Problem 2E: If a binomial experiment has probability p success, then the probability of failure is...
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Example 5.28 (Uniform Distribution). Let S be a set containing N elements; for example, S could be the set S = {0, 1,..., N-1}. Let X be a random variable satisfying 1 fx(j) = Pr(X= j) = N This random variable X is said to be uniformly distributed or to have uniform density, since each of the outcomes in S is equally likely. Example 5.29 (Binomial Distribution). Suppose that an experiment has two outcomes, success or failure. Let p denote the probability of success. The experiment is performed n times and the random variable X records the number of successes. The sample space 2 consists of all binary strings w = bib2.bn of length n, where b; = 0 if the i'th experiment is a failure and bį = 1 if the i'th experiment is a success. Then the value of the random variable X at w is simply X (w) = b₁+b₂+ +bn, which is the number of successes. Using the random variable X, we can express the probability of a single event was Pr({w})= px (w) (1 - p)n-X(w). (Do you see why this is the correct formula?) This allows us to compute the probability of exactly k successes as fx (k)= Pr(X= k) Σ WEN, X(w)=k = = = Pr({w}) = if j = S, if j & S. Σ p*(w)(1 −p)"- WEN, X(w)=k Σp (1-p)n-k WEN, X(w)=k = # {w EN: X(w) = k}pk (1-p)n-k () ₁² (1) )n-X (w) fx (k) = (7) ₁² (1– Here the last line follows from the fact that there are (2) ways to select exactly k of the n experiments to be successes. The function is called the binomial density function. - p)n-k (5.23)
Transcribed Image Text:Example 5.28 (Uniform Distribution). Let S be a set containing N elements; for example, S could be the set S = {0, 1,..., N-1}. Let X be a random variable satisfying 1 fx(j) = Pr(X= j) = N This random variable X is said to be uniformly distributed or to have uniform density, since each of the outcomes in S is equally likely. Example 5.29 (Binomial Distribution). Suppose that an experiment has two outcomes, success or failure. Let p denote the probability of success. The experiment is performed n times and the random variable X records the number of successes. The sample space 2 consists of all binary strings w = bib2.bn of length n, where b; = 0 if the i'th experiment is a failure and bį = 1 if the i'th experiment is a success. Then the value of the random variable X at w is simply X (w) = b₁+b₂+ +bn, which is the number of successes. Using the random variable X, we can express the probability of a single event was Pr({w})= px (w) (1 - p)n-X(w). (Do you see why this is the correct formula?) This allows us to compute the probability of exactly k successes as fx (k)= Pr(X= k) Σ WEN, X(w)=k = = = Pr({w}) = if j = S, if j & S. Σ p*(w)(1 −p)"- WEN, X(w)=k Σp (1-p)n-k WEN, X(w)=k = # {w EN: X(w) = k}pk (1-p)n-k () ₁² (1) )n-X (w) fx (k) = (7) ₁² (1– Here the last line follows from the fact that there are (2) ways to select exactly k of the n experiments to be successes. The function is called the binomial density function. - p)n-k (5.23)
5.34. In each case, compute the expectation of the random variable X. (a) The values of X are uniformly distributed on the set {0, 1, 2,..., N-1}. (See Example 5.28.) (b) The values of X are uniformly distributed on the set {1,2,..., N}. (c) The values of X are uniformly distributed on the set {1, 3, 7, 11, 19, 23}. (d) X is a random variable with a binomial density function; see formula (5.23) in Example 5.29 on page 240.
Transcribed Image Text:5.34. In each case, compute the expectation of the random variable X. (a) The values of X are uniformly distributed on the set {0, 1, 2,..., N-1}. (See Example 5.28.) (b) The values of X are uniformly distributed on the set {1,2,..., N}. (c) The values of X are uniformly distributed on the set {1, 3, 7, 11, 19, 23}. (d) X is a random variable with a binomial density function; see formula (5.23) in Example 5.29 on page 240.
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