Discrete and Random Variables

Question 1

Probability distribution

random variable

probability

discrete random variable

continuous random variables

Answer 1

A random variable is a variable whose value, which cannot be known beforehand, is determined by the outcome of an experiment

The probability of its occurrence can be predicted

Discrete random variables are often the result of a counting process and are often whole numbers

Continuous random variables are often the result of measurement and are usually from the set of real numbers

Question 2

Probability table

probability table description

all assigned variables

Answer 2

A probability distribution or probability function is a table which shows the probabilities associated with each value of a random variable

The sum of all assigned must be equal to 1
Σ P(X = x) = 1

Question 3

Expectation

probability distribution and expected values

note

expected value notation

expected value formula

note

Answer 3

A probability distribution can have a mean or average value, this is known as the expected value or expectation

Note: the expected value need not to be one of the possible values

The symbol for the expected value of the distribution of x is written E(X)
E(X) = µ (mu, the same symbol as the population mean)

E(X) = µ = ∑ xᵢPᵢ

You cannot score 3.5 on a dice in one throw, but if you throw the dice 10 times and added the scores, you would expect to get the total around ≈ 35 (3.5 x 10)

Question 4

Variance of a random variable

description

formula and notation

note

Answer 4

The expected value or mean of a probability function of a random variable gives the measure of the average or central tendency, but no indication of how spread the values are likely to be

The standard deviation (σ) or variance (σ²) of a probability distribution is the measure of the spread of likely values of a random variable

The variance of the discrete random variable X with
E(X) = µ is given by:
σ² = Var(X)
= E(X - µ)² = ∑ (xᵢ - µ)² x P(xᵢ)

The standard deviation of a random variable is σ, the square root of Var(X)
Var(X) = σ²
Sd(X) = √σ² = σ

Question 5

Alternative formula for the variance of a random variable

Answer 5

Var(X) = E(X²) - [E(X)]²