Question

Probability function of the random variable X is

f(x) = k/(2^x) , where x = 0,1,2,...

a. Compute k

b. Compute P(X ≥ 6)

c. Compute P(1 ≤ X ≤ 6)

Answer 1

a) $1=\sum\limits_{x=0}^{\infty}P(X=x)=\sum\limits_{x=0}^{\infty}\frac{k}{2^x}=2k$, hence, $k=1/2$.

b) $P(X \geq 6)=\sum\limits_{x=6}^{\infty}P(X=x)=\sum\limits_{x=6}^{\infty}\frac{1}{2^{x+1}}=\frac{2^{-7}}{1-\frac{1}{2}}=\frac{1}{64}$

c) $P(1 ≤ X ≤ 6)=\sum\limits_{x=1}^{6}P(X=x)=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}=\frac{63}{128}$