Discrete Random Variables - Basics

What is a random variable?

A random variable, usually represented by a capital letter, represents the outcome of a random process or experiment. Random variables can take different values, each with their own associated probability.

What are independent random variables?

Two random variables are independent of each other if the outcome of one does not effect the outcome of the other.

What is the expected value for two independent OR dependent random variables?

For both independent and dependent random variables, the linearity of expectation law always holds, which means that:
E[X+Y] = E[X] + E[Y]
aka. remember this isn’t about probability any more.
The expected value (or MEAN) of a random variable is a measure of the average outcome you would expect. We will learn how to calculate this later - for now, we are just thinking about the relationships between the expected values of multiple independent/dependent random variables.

What is the variance for an independent random variable?

Variance is a measure of how spread the values of a random variable are around its mean. Since the outcome of one independent random variable does not effect the outcomes of the others, the total variance of multiple independent random variables can just be added together, because they don’t interfere with each other.
(X1+X2+⋯+Xn)=V(X1)+V(X2)+⋯+V(Xn)

What is a discrete random variable?

A discrete random variable is a type of random variable that can only take a finite number of distinguishable values, like the outcomes of a coin toss.

What are the two laws that govern discrete random variables?

1. Functions of random variables are, in of themselves, random variables.

2. Law of the Unconscious Statistician - Even if we don’t know the probability distribution of a random variable function, we can find the expected value of that function. E(g(X))=∑x g(x)×P(X=x)

A little explanation on why the LOTUS is cool:

So, for something like a dice, this law is really cool. Because the “probability distribution” across all the values of the random variable of rolling that dice are the same (1/6), we can use LOTUS to find the most likely result.
1. Apply the function to each possible result of the random variable
2. Multiply each result by the probability of that result of the random variable (so for a dice, this would be 1/6 for results 1 through 6).
3. Add them all up!

What is the expected value (mean) of a discrete random variable?

μ=E(X)= ∑x [xf(x)]
Where x is all the possible values that X can take, and f(x) is the PMF, being the weighted probability of each outcome of x - so P(X = x). You just add them all up, basically.

What is the variance of a discrete random variable?

σ^2 = V(X) = E[(X-μ)^2] = ∑x ((x-μ)^2 f(x)) = ∑x (x^2 f(x)-μ^2)

What is the standard deviation of a discrete random variable?

σ=√(σ^2)