Question

 If x has the probability function 𝑓(𝑥) = 𝑘/2^x for x=0,1,2,……… then determine the followings

a) Vale of k

b) P(x≥ 4)

Answer 1

By definition,

$\displaystyle \sum_{x=0}^\infty \frac{k}{2^x} = 1 = k\sum_{x=0}^\infty \frac{1}{2^x} = k \cdot 2$

$k=1/2$

$\displaystyle P(X \geq 4) = 1 - P(X < 4) = 1 - \frac{1}{2}\sum_{x=0}^3 \frac{1}{2^x}= 1 - \frac{1}{2} \cdot \frac{15}{8} = 1 - \frac{15}{16} = \frac{1}{16}$

Answer: $k=1/2; P(X\geq 4) = 1/16.$