Question

Show that

f(x)=(1/2)^(x+1) , x = 0, 1, 2, 3, 4, 5,...

is a valid PMF for a discrete random variable. Also find

out its CDF.

Answer 1

a.

$$\displaystyle\sum_{x=0}^{\infin}(\dfrac{1}{2})^{x+1}=\dfrac{1/2}{1-1/2}=1, True$$

Therefore $f(x)(\dfrac{1}{2})^{x+1}, x=0,1,2,3,4,5,...$ is a valid PMF for a discrete random variable.

b.

$$F(x)=\displaystyle\sum_{y\le x}^{x}f(y)=\displaystyle\sum_{y=0}^{x}(\dfrac{1}{2})^{y+1}=\dfrac{1}{2}\displaystyle\sum_{y=0}^{x}(\dfrac{1}{2})^{y}$$

$$=\dfrac{1}{2}(\dfrac{1-(1/2)^{x+1}}{1-1/2})=1-(\dfrac{1}{2})^{x+1}$$

$$F(x)= \begin{cases} 0&x<0 \\ 1-(\dfrac{1}{2})^{x+1}&x=0,1,2,3,4,5,... \end{cases}$$