# Discrete random variables and PMF

## Homework Statement

A discrete random variable X has the following PMF

x | 1 | 2 | 3 | 4 | 5 |
p(x)|1-a|1-2a|0.2| a | 0.5a|

What are the values of "a" that are allowed in this PMF?
For the allowed values, compute the expected value and the standard deviation of the variable X.

## Homework Equations

Each p(xi) >= 0 and the sum of all p(xi) from i = 1 to i = 5 is equal to 1
E(x) = x1(p(x1)) + x2(p(x2)) + x3(p(x3)) + x4(p(x4)) + x5(p(x5))
E(x^2) = x1^2(p(x1)) + x2^2(p(x2)) + x3^2(p(x3)) + x4^2(p(x4)) + x5^2(p(x5))
V(x) = E(x^2) - [E(x)]^2
SD (standard deviation) = sq. root of V(X)

## The Attempt at a Solution

(1-a) + (1-2a) + 0.2 + a + 0.5a = 1
2.2 - 1.5a = 1 => a = 0.8

E(x) = (1)(1-(.8)) + (2)(1-2(.8)) + (3)(0.2) + (4)(0.8) + (5)(.5 * .8) = 4.8
E(x^2) = (1)^2(1-(.8)) + (2)^2(1-2(.8)) + (3)^2(0.2) + (4)^2(0.8) + (5)^2(.5 * .8) = 22.4
V(x) = 22.4 - [4.8]^2 => V(x) = -0.64, which is not possible since variance is nonnegative or else we would need to take the sq. root of a negative number to get SD, which can only be a real, positive value......

Any help would be greatly appreciated since I need to turn in this HW by Monday!

-Red88