Let the distribution of X be ?2(r).(a) Find the point at

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

ISBN: 9780321923271 41

Solution for problem 15E Chapter 3.2

Probability and Statistical Inference | 9th Edition

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Problem 15E

Let the distribution of X be χ2(r).

(a) Find the point at which the pdf of X attains its maximum when r ≥ 2. This is the mode of a χ2(r) distribution.

(b) Find the points of inflection for the pdf of X when r ≥ 4.

Step-by-Step Solution:

Answer:

Step 1 of 4:

Let the distribution of ‘X’ be ${\chi{}}_{\ }^{2}(r).$

The probability density function of ${\chi{}}_{\ }^{2}$ distribution with ‘r’ degrees of freedom is

$f(x)=\frac{1}{\Gamma{}(\frac{r}{2}){\ 2}^{\frac{r}{2}}}x{\ }^{\frac{r}{2}-1\ }e{\ }^{-\frac{x}{2}}\$

Step 2 of 4:

a). Now we have to find the derivative of the pdf of the ${\chi{}}_{\ }^{2}$ distribution.

$\frac{d}{dx}f(x)$ = $\frac{d}{dx}(\frac{1}{\Gamma{}(\frac{r}{2}){\ 2}^{\frac{r}{2}}}x{\ }^{\frac{r}{2}-1\ }e{\ }^{-\frac{x}{2}}\ )$

= $\frac{1}{\Gamma{}(\frac{r}{2})2{\ }^{\frac{r}{2}}}\frac{d}{dx}(x{\ }^{\frac{r}{2}-1}e{\ }^{-\frac{x}{2}})$

= $\frac{1}{\Gamma{}(\frac{r}{2})\ 2{\ }^{\frac{r}{2}}}$ (( $\frac{r}{2}-1)x{\ }^{\frac{r}{2}-2\ e{\ }^{-\frac{x}{2}\ \ }}-\ \frac{1}{2}x{\ }^{\frac{r}{2}-1}e{\ }^{\frac{-x}{2}})$

f $'$ (x) = $\frac{x{\ }^{\frac{r}{2}-2\ e{\ }^{\frac{-x}{2}}}}{\Gamma{}(\frac{r}{2})\ 2{\ }^{\frac{r}{2}}\ }(\frac{r}{2}-1-\frac{x}{2})$

Equating the derivative to zero to find the extreme point, that is $\frac{d}{dx}f(x)$ = 0.

$\frac{x{\ }^{\frac{r}{2}-2\ e{\ }^{\frac{-x}{2}}}}{\Gamma{}(\frac{r}{2})\ 2{\ }^{\frac{r}{2}}\ }(\frac{r}{2}-1-\frac{x}{2})$ = 0

$\Rightarrow{}(\frac{r}{2}-1-\frac{x}{2})\ =\ 0$

$\Rightarrow{}$ $\frac{r-2-x}{2}=0$

$\Rightarrow{}$ $r-2-x$ = 0

Therefore, x = r - 2.

We know that f(0) = 0 , and ( $r-2-x$ ) can ever be zero at x > 0, so the critical point is x = r - 2. Here the pdf attains is maximum for x = r - 2, where ‘r’ is the degrees of freedom.

Therefore, the mode of distribution is x = r - 2.

Step 3 of 4