1 to explore the shape of the pmf we examine the

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Unformatted text preview: n of p in Figure 2.4. It is symmetric about 1/2, and achieves its maximum value, 1/4, at p = 1/2. Figure 2.4: The variance of a Bernoulli random variable versus p. 2.4.4 Binomial distribution Suppose n independent Bernoulli trials are conducted, each resulting in a one with probability p and a zero with probability 1 − p. Let X denote the total number of ones occurring in the n trials. Any particular outcome with k ones and n − k zeros, such as 11010101, if n = 8 and k = 5, has 34 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES probability pk (1 − p)n−k . Since there are n k such outcomes, we find that the pmf of X is nk p (1 − p)n−k k pX (k ) = for 0 ≤ k ≤ n. The distribution of X is called the binomial distribution with parameters n and p. Figure 2.5 shows the binomial pmf for n = 24 and p = 1/3. Since we just derived this pmf, we know that it sums Figure 2.5: The pmf of a binomial random variable with n = 24 and p = 1/3. to one. We will double check that fact using a series expansion. Reca...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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