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Chapter 5: Q. 5.14 (page 213)

Let Xbe a uniform (0,1)random variable. Compute role="math" localid="1646717640777" EXnby using Proposition 2.1, and then check the result by using the definition of expectation.

Short Answer

Expert verified

The required answer isEXn=1n+1.

Step by step solution

01

Step 1. Given Information.

Here, it is given that: Xis a uniform0,1random variable.

02

Step 2. Write the probability density function of uniform distribution.

Let Xbe a random variable and follows a uniform distribution with parameters a=0andb=1.

The probability density function of uniform distribution is f(x)=1b-a;a<x<b

03

Step 3. Calculate EX from the expectation definition for uniform distribution Ua,b.

EX=鈭�abx1b-adx=鈭�01x11-0dx=鈭�01xdx

04

Step 4. Compute EXn by using definition of expectation.

EXn=鈭�01xdx=xn+1n+101=1n+1n+1-0n+1n+!=1n+1鈭�EXn=1n+1

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Most popular questions from this chapter

The number of minutes of playing time of a certain high school basketball player in a randomly chosen game is a random variable whose probability density function is given in the following figure:

Find the probability that the player plays

(a) more than 15minutes;

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Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.
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