Module 8: Functions of Random Variables, Expectation, and Variance

Question 1

The PMF of a random variable X provides us with several numbers, namely…

Answer 1

the probabiltiies of all possibilities of X. It is often desireable to summarize this information in a signle representative number.

Question 2

what is the def of the expected value of a discrete random varaible X with a PmF P({X = x}) = p(x) is given by:

Answer 2

E(X) = μ = SUM(x * p(x))

Question 3

what is the thrm for Expected Value Rule for Functions of Random Variables?

Answer 3

Let X be a random variable with PMF p(x) and let g(X) be real-valued function of X. Then, the expected value of the random variable Y = g(X) is given by:

E(Y) = E(g(X) = SUM(g(x) * p(x))

Question 4

Whats the def of Variance for a random variable X and the standard deviation?

Answer 4

If X is a random variable with expected value μ, then the variance of X is given by:

Var(X) = σ^2 = E[(X -μ)^2]

and the standard deviation is given by the square root of the variance:

σ = sqrt(E[(X - μ)^2])

Question 5

what is the thrm for calcing Var(X)?

Answer 5

The variance of a random variable X can be calculated using the formula:

Var(X) = σ^2 E(X^2) - (E(X))^2

Question 6

what does E(X^2) mean?

Answer 6

We note that E(X^2) is called the second moment of X where more generally, we define the nth moment as the expected value of the random variable X^n:

E(X^n) = SUM(x^n * p(x))