Question

The Probability Density Function (PDF) of

$f\left(x\right)isgivenas;f_x\left(x\right)=\frac{1}{20}\left[u\right(x+10)-u(x-10\left)\right]$ and X and y are related as Then,

• A. $P(y>1)=0$
• B. $f_{Y\left(y\right)}=\frac{1}{2}\delta (y+1)-\frac{1}{2}\delta \left(y\right)+\frac{1}{4}\delta (y-1)$
• C. $F_Y\left(0\right)=\frac{1}{2}$
• D. $P(Y\le 0)=P(Y\ge 0)$

Answer 1

Answer: (A), (D)

$P(Y\le 0)=P(Y\ge 0)$

$Y\in \{-1,0,1\}$

$P(Y=-1)=P(-5\le X\le 0)=\frac{1}{4}$

$P(Y=0)=P(X<-5)+P(X>5)$

$=\frac{4}{20}+\frac{5}{20}=\frac{1}{2}$



$P(Y=1)=P(0\le X\le 5)=\frac{1}{4}$

Hence, $f_Y\left(y\right)=\frac{1}{4}\delta (y+1)+\frac{1}{2}\delta \left(y\right)+\frac{1}{4}\delta (y-1)$

$F_Y\left(0\right)=P(Y\le 0)=\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$