Random Variables and Probability Distributions: Key Concepts and Formulas

Random Variables and Probability Distributions

Random Variables

A random variable is a variable that assumes any of several different numeric values as a result of some random event. Random variables are typically denoted by a capital letter such as X.

• Discrete Random Variable: A random variable that can take on only a finite number of distinct outcomes. Example: Number of heads in 10 coin tosses.

• Continuous Random Variable: A random variable that can take any numeric value within an interval or collection of intervals. Example: Heights of students in a class.

Probability Model

A probability model is a function that assigns a probability to each possible outcome of a random variable. For a discrete random variable X, the probability model is often written as P(X = x) for each possible value x.

Cumulative Probability Distribution

For a random variable X, the cumulative probability distribution gives the probability that X is less than or equal to a certain value. It is denoted as P(X ≤ x).

Expected Value (Mean) of a Random Variable

The expected value (or mean) of a random variable is its theoretical long-run average value, calculated as a weighted sum of all possible values, weighted by their probabilities.

• For a discrete random variable X with possible values x_i and probabilities p_i:

$E(X) = \mu_X = \sum x_i p_i$

Variance and Standard Deviation of a Random Variable

• Variance: The variance of a random variable is the expected value of the squared deviation from the mean. It measures the spread of the distribution.

• For a discrete random variable X:

$Var(X) = \sigma_X^2 = \sum (x_i - \mu_X)^2 p_i$

• Standard Deviation: The standard deviation is the square root of the variance.

$\sigma_X = \sqrt{Var(X)}$

Linear Transformations of Random Variables

• If a random variable X is multiplied by a constant a and/or a constant b is added, the expected value and variance change as follows:

$E(aX + b) = aE(X) + b$

$Var(aX + b) = a^2 Var(X)$

Addition Rule for Expected Values and Variances

• Expected Value: For any two random variables X and Y:

$E(X + Y) = E(X) + E(Y)$

• Variance (if X and Y are independent):

$Var(X + Y) = Var(X) + Var(Y)$

• Standard Deviation:

$SD(X + Y) = \sqrt{Var(X) + Var(Y)}$

Example

• Suppose X is the number of heads in two coin tosses. The possible values are 0, 1, and 2, with probabilities 0.25, 0.5, and 0.25, respectively.

• Expected value: $E(X) = 0 \times 0.25 + 1 \times 0.5 + 2 \times 0.25 = 1$

• Variance: $Var(X) = (0-1)^2 \times 0.25 + (1-1)^2 \times 0.5 + (2-1)^2 \times 0.25 = 0.5$

• Standard deviation: $SD(X) = \sqrt{0.5} \approx 0.707$

Additional info: The notes above summarize foundational concepts for discrete and continuous random variables, probability models, and the calculation of expected value, variance, and standard deviation. These are essential for understanding probability distributions and inferential statistics.