Unit 3: Random Variables and Probability Distributions

1. Defining Random Variables

A Random Variable (R.V.) is a rule that assigns a numerical value to each outcome in a sample space.

• Discrete Random Variable: Can take only finite or countably infinite values (e.g., the number of customers entering a bank).
• Continuous Random Variable: Can take any value within a specified range (e.g., the time it takes for an investment to mature).

2. PMF, PDF, and Cumulative Probability Function

These functions describe how probabilities are assigned to the values of a random variable.

Probability Mass Function (PMF)

Used for discrete random variables. It gives the probability that a discrete R.V. is exactly equal to some value.

Probability Density Function (PDF)

Used for continuous random variables. The probability of the R.V. falling within a particular range is given by the area under the PDF curve over that range.

Cumulative Distribution Function (CDF)

Gives the probability that a random variable X will take a value less than or equal to x: P(X ≤ x).

3. Mathematical Expectation and Theorems

Mathematical Expectation is the "long-run average" or expected value of a random variable.

Formula: E(X) = Σ [x * P(x)] for discrete variables.

Theorems on Expectation:

• E(c) = c: The expected value of a constant is the constant itself.
• E(aX + b) = aE(X) + b: Expectation is a linear operator.
• E(X + Y) = E(X) + E(Y): The expectation of a sum is the sum of expectations.

4. Discrete Distributions: Binomial and Poisson

Binomial Distribution

Used when there are exactly two mutually exclusive outcomes of a trial (e.g., success/failure).

• Parameters: n (number of trials) and p (probability of success).
• Properties: Mean = np; Variance = npq.

Poisson Distribution

Used for events that occur rarely over a fixed interval of time or space.

• Parameters: λ (average number of events in the interval).
• Unique Property: Mean = Variance = λ.

5. Continuous Distribution: Normal Distribution

The Normal Distribution is the most important distribution in economics and statistics because many natural and economic phenomena follow it.

Properties:

• Symmetry: It is bell-shaped and symmetric about the mean.
• Equality: Mean = Median = Mode.
• Area Rule: Approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.