Chapter 7: Q7.20 (page 364)

Let $X$ be a nonnegative random variable having a distribution function $F$ . Show that if $\bar{F} (x) = 1 - F (x)$ , then
$E [X^{n}] = \int_{0}^{\infty} x^{n - 1} \bar{F} (x) d x$
Hint: Start with the identity

$X^{n} = n \int_{0}^{X} x^{n - 1} d x$
$= n \int_{0}^{\infty} x^{n - 1} I_{X} (x) d x$
where
$I_{x} (x) = \{\begin{cases} 1 \\ 0 \end{cases}$ if $x < X$ otherwise

Short Answer

Expert verified

$= \bar{F} (x)$
Hence $E (X^{n}) = n \int_{0}^{\infty} x^{n - 1} \bar{F} (x) d x$

Step by step solution

Concept Introduction

Let $X$ be a Non-negative Random variable having distribution function $F$ .
With $\bar{F} (x) = 1 - F (x)$

:Explanation

Let $X$ be a Non-negative Random variable having distribution function $F$ .
With $\bar{F} (x) = 1 - F (x)$

:Explanation

$X^{n} = n \int_{0}^{x} x^{n - 1} d x$

Step 4:Explanation

$= n \int_{0}^{\infty} x^{n - 1} I_{x} (x) d x$ $= n \int_{0}^{\infty} x^{n - 1} I_{x} (x) d x$ $= n \int_{0}^{\infty} x^{n - 1} I_{x} (x) d x$ $= n \int_{0}^{\infty} x^{n - 1} I_{x} (x) d x$

Step 5:Explanation

Where $I_{x} (x) = \{\begin{cases} 1 \\ 0 \end{cases}$ if $x < X$ $x < X$
otherwise

Step 6:Explanation

$∴ E [I_{X} (x)] = 1 . P [x < X]$

Step 7:Explanation

$= P [X > x]$

Step 8:Explanation

$= 1 - P [X \leq x]$ $= 1 - P [X \leq x]$ $= 1 - P [X \leq x]$
$= 1 - F (x)$

Final Answer

$= \bar{F} (x)$
Hence $E (X^{n}) = n \int_{0}^{\infty} x^{n - 1} \bar{F} (x) d x$

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Short Answer

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Concept Introduction

:Explanation

:Explanation

Step 4:Explanation

Step 5:Explanation

Step 6:Explanation

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Step 8:Explanation

Final Answer

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Let Xbe a nonnegative random variable having a distribution function F. Show that if F¯(x)=1-F(x), thenEXn=∫0∞xn-1F¯(x)dxHint: Start with the identityXn=n∫0Xxn-1dx=n∫0∞xn-1IX(x)dxwhereIx(x)=10ifx<X otherwise

Short Answer

Step by step solution

Concept Introduction

:Explanation

:Explanation

Step 4:Explanation

Step 5:Explanation

Step 6:Explanation

Step 7:Explanation

Step 8:Explanation

Final Answer

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Most popular questions from this chapter

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