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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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- Spring '08
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