An Introduction to Mathematical Statistics and Its Applications (6th Edition)

Published by Pearson
ISBN 10: 0-13411-421-3
ISBN 13: 978-0-13411-421-7

Chapter 3 Random Variables - 3.3 Discrete Random Variables - Questions - Page 127: 14

Answer

$\color{blue}{f_X(x) = x/21, x=0,\ldots, 6}.$

Work Step by Step

For $X=0$, $\begin{align*} f_X(0) &= P(X\le 0) - P(X\lt 0) \\ &= F(0) - 0 & [\text{since}\ F_X(x)\ne 0,\ \text{only for}\ x=0,1,\ldots, 6] \\ &= \frac{0(0+1)}{42} \\ f_X(0) &= 0. \end{align*}$ For $X=1,\ldots, 6,$ $\begin{align*} f_X(x) &= P(X=x) \\ &= P(X\le x) - P(X\lt x),\ x=1,\ldots,6 \\ &= P(X\le x) - P(X\le x-1) & [\text{since}\ F_X(x)\ne 0,\text{only for}\ x=0,1,\ldots, 6] \\ &= F(x) -F(x-1) \\ &= (x)(x+1)/42 - (x-1)(x)/42 \\ &= (x)((x+1)-(x-1))/42 \\ &= (x)(2)/42 \\ f_X(x) &= x/21 \end{align*}$ Since $f_X(0) = 0/21 =0 ,$ we can then define $\color{blue}{f_X(x) = x/21, x=0,\ldots, 6}.$
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