Let x denote the number of biased die with the following probability mass function p(x) X: 1 2 3 4 5 6 P(x): 1/6 1/6 1/5 y 1/5 1/6 a) find the value of y b) find the cdf of x

a)

$\sum_ip(x_i)=1$

$1/6+1/6+1/5+y+1/5+1/6=1$

$y=1/10$

b)

$F(1) =P(X\le 1)=P(1)= 1/6$

$F(2) =P(X\le 2)=P(1)+P(2)= 1/3$

$F(3) =P(X\le 3)=P(1)+P(2)+P(3)= 8/15$

$F(4) =P(X\le4)=P(1)+P(2)+P(3)+P(4)$

$= 19/30$

$F(5) =P(X\le5)=P(1)+P(2)+P(3)+P(4)$

$+P(5)=5/6$

$F(6)=P(X\le6)=P(1)+P(2)+P(3)+P(4)$

$+P(X=5)+P(X=6)=1$

The cdf is

$F(x)=\begin{cases} 0 &x<1 \\ 1/6 &1\le x< 2 \\ 1/3 &2\le x< 3 \\ 8/15 & 3\le x< 4 \\ 19/30 & 4\le x< 5 \\ 5/6 &5\le x< 6 \\ 1 & x\ge 6 \\ \end{cases}$