Question

"Let X be a binomial random variable with parameters n and p. Show that E[X + 1] = (n + 1)p / (1 - (1 - p)^(n + 1))."

Answer 1

To calculate E[X + 1] for a binomial random variable with parameters n and p, we first find the expected value of X and then add 1.

Step-by-step explanation:

To show that E[X + 1] = (n + 1)p / (1 - (1 - p)^(n + 1)), we need to calculate the expected value of X + 1. The expected value of a random variable is the sum of the products of each possible value of the variable and its corresponding probability.

First, let's calculate the expected value of X:

• The expected value of X, denoted as E[X], is given by E[X] = np.

Next, we need to calculate the expected value of X + 1:

• We can rewrite X + 1 as X + 1 = X + 1 * 1.
• Using the linearity of expectation, we have E[X + 1] = E[X] + E[1].
• The expected value of 1 is 1, so we have E[X + 1] = E[X] + 1.
• Substituting E[X] = np, we get E[X + 1] = np + 1.

Finally, rearranging the equation, we have E[X + 1] = (n + 1)p.

So, the expression E[X + 1] = (n + 1)p / (1 - (1 - p)^(n + 1)) is correct.