Question

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

Answer 1

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

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