Answer to Question #242102 in Statistics and Probability for jack

Question

Probability function of the random variable X is:

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

a. Compute k.

b. Compute P(X ≥ 6).

c. Compute P(1 ≤ X ≤ 6).

Answer 1

a.

$$\displaystyle\sum_{x=0}^{\infin}\dfrac{k}{2^x}=k(\dfrac{1}{1-\dfrac{1}{2}})=2k=1=>k=\dfrac{1}{2}$$

b.

$$P(X\geq6)=1-P(X<6)$$

$$=1-(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64})$$

$$=1-\dfrac{63}{64}=\dfrac{1}{64}$$

c.

$$P(1\leq X\leq6)=\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64}+\dfrac{1}{128}$$

$$=\dfrac{63}{128}$$