1 to explore the shape of the pmf we examine the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online