Question

Let X be a discrete random variable with p.m.f given by

P(x) = 1/2x ' x = 1,2,3, ... (a) Find F(x) and use it to compute P(X > 4)
(b) Find P(2 < X <=5)

Answer 1

It looks like the given PMF is

`P(X=x) = \begin{cases}1/2^x & \text{if }x\in\{1,2,3,\ldots\} \\ 0 & \text{otherwise}\end{cases}`

Then we have

`P(X\le x) = \displaystyle \sum_(i=1)^x P(X=i) \\\\ ~~~~~~~~~~~~~~ = \sum_(i=1)^x \frac1{2^i} \\\\ ~~~~~~~~~~~~~~ = \frac12 + \frac1{2^2} + \frac1{2^3} + \cdots + \frac1{2^x}`

Multiply both sides by 1/2.

`\frac12 P(X\le x) = \frac1{2^2} + \frac1{2^3} + \frac1{2^4} + \cdots + \frac1{2^(x+1)}`

Subtract this from
`P(X\le x)`.

`\frac12 P(X\le x) = \frac12 - \frac1{2^(x+1)} \\\\ P(X\le x) = 1 - \frac1{2^x}`

So the CDF is

`F(x) = \begin{cases} 0 & \text{if } x < 1 \\\\ 1 - 1/2^x & \text{if } x \ge 1 \end{cases}`

(a) Using the CDF and probability of the complement,

`P(X > 4) = 1 - P(X \le 4) \\\\ ~~~~~~~~~~~~~~= 1 - F(4) \\\\ ~~~~~~~~~~~~~~= 1 - \frac1{2^4} = \boxed{(15)/(16)}`

(b) We can use the CDF again here.

`P(2 < X \le 5) = P(X \le 5) - P(X \le 2) \\\\ ~~~~~~~~~~~~~~~~~~~~= F(5) - F(2) \\\\ ~~~~~~~~~~~~~~~~~~~~= \left(1 - \frac1{2^5}\right) - \left(1 - \frac1{2^2}\right) = \boxed{\frac7{32}}`