Statistics 1 - Discrete random variables

Question 1

What is a random variable?

Answer 1

A variable whose value depends on a random event.

Question 2

What is a discrete variable?

Answer 2

A variable that can only take certain numerical values.

Question 3

How can the expected value of a discrete random variable be discerned?

Answer 3

Take a set of observations from a discrete random variable. Then find the mean of those observations. As the number of observations increases, the mean of the observations will get closer to the expected value of the discrete random variable. As such the expected value is sometimes referred to as the mean and denoted as μ.

Question 4

How can the expected value of a discrete random variable, X, be calculated?

Answer 4

E(𝑋)=ΣxP(𝑋=𝑥)
Where E(𝑋) is the expected value and 𝑥P(𝑋=𝑥) is the value of 𝑥 multiplied by the probability of the variable 𝑋 taking the value 𝑥.

Question 5

How can the expected value of X² be calculated, where X is a discrete random variable?

Answer 5

If 𝑋 is a discrete random variable then so too is 𝑋², thus; E(𝑋²)=Σx²P(𝑋=𝑥)
Where E(𝑋) is the expected value and 𝑥P(𝑋=𝑥) is the value of 𝑥 multiplied by the probability of the variable 𝑋 taking the value 𝑥.

Question 6

How can the variance of a discrete random variable, X, be calculated?

Answer 6

Var(𝑋)=E((𝑋-E(𝑋))²)=E(𝑋²)-(E(𝑋))²
Where Var(𝑋) is the variance of 𝑋, E(𝑋) is the expected value and E(𝑋²) is the expected value of 𝑋².

Question 7

How would the expected value of a function of X be calculated?

Answer 7

Generally; E(g(𝑋))=Σg(𝑥)P(𝑋=𝑥)
As such the following rules are true;
If 𝑋 is a random variable and a and b are constants, then E(𝑎𝑋+𝑏)=𝑎E(𝑋)+𝑏

Question 8

How would the variance of a function of X be calculated?

Answer 8

If 𝑋 and 𝑌 are random variables, then E(𝑋+𝑌)=E(𝑋)+E(𝑌)

If 𝑋 is a random variable and 𝑎 and 𝑏 are constants then Var(𝑎𝑋+𝑏)=𝑎²Var(𝑋)