Question

In: Math

1. Suppose that X is a Geometric r.v. and its pmf is f(x; p) = p(1...

1. Suppose that X is a Geometric r.v. and its pmf is

f(x; p) = p(1 − p) x−1 , x = 1, 2, . . . and 0 ≤ p ≤ 1.

a. Calculate the mean of X.

b. Calculate the median of X.

c. Check and show whether the median is smaller than mean.

Solutions

Expert Solution

a) The mean here is computed as:

Multiplying both sides by (1-p), we get here:

Subtracting the last equation from the second last one, we get here:

Cancelling out p from both sides, we get here:

therefore 1/p is the required mean of X here.

b) Let the median be M, then we have here:

This is the required median for X here.

c) Let us define a function K as:

K = Mean - Median

Differentiating K with respect to p, we get here:

As the first derivate of K is less than 0, therefore this is a decreasing function in p.

For p = 0.1, we have:

Therefore for p = 0.1, Mean > Median

For p = 0.99,

Again for p = 0.99, Mean > Median

Therefore we can conclude here that the mean is greater than the median.

milcah answered 8 months ago

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