Question

Determine the expectation, variance, standard deviation, and cumulative distribution function (c.d.f.) for each of the following:

(a)

x	0	1	2	3	4
f(x)	0.44	0.36	0.15	0.04	0.01

(b) f(x) = x 10 for x = 1, 2, 3, 4

(c) f(x) = 2 5x2−30x+45 for x = 1, 2, 4, 5

(d) A random variable that represents the outcome of rolling one (fair) die.

(e) A random variable that represents the outcome of rolling two (fair) dice.

Answer 1

The CDF is obtained by adding the probabilities of any value upto that point.

The expected value is defined as -

$E[X] = \sum x_{i} f(x_{i})$

Variance is given as -

$Var(X) = E[X^{2}] - (E[X])^2$

and standard deviation is just the square root of variance.

Also note that part (c) is not a probability distribution function as the probabilities don't add up to 1. So, we can't find CDF, mean or variance

In part (e), what is the event you are interested in? Is it sum of two values? Product? Please ask this part separately as you are allowed maximum of 4 parts in a question.