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Statistics / Introduction to Probability 1 / Chapter 4 / Problem 59

Introduction to Probability | 1st Edition | ISBN: 9781466575578 | Authors: Joseph K. Blitzstein, Jessica Hwang

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Let X Geom(p) and let t be a constant. Find E(etX), as a function of t (this is known as

Chapter 4, Problem 59

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QUESTION:

Let X Geom(p) and let t be a constant. Find E(etX), as a function of t (this is known as the moment generating function; we will see in Chapter 6 how this function is useful).

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QUESTION:

Let X Geom(p) and let t be a constant. Find E(etX), as a function of t (this is known as the moment generating function; we will see in Chapter 6 how this function is useful).

ANSWER:

Step 1 of 2

According to the law of unconscious statistics, for a discrete random variable and a real valued function of single variable,

The probability density function for a random variable is

,

Now, we need to find , is any constant.

Here, . Then,

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