1. Random Variable (RV)

A random variable is a variable whose values are outcomes of a random experiment.

• Usually denoted by $X, Y, Z$
• It assigns a numerical value to each outcome in a sample space.

Example:

• Toss a coin: $X = 1$ if Head, $X = 0$ if Tail
• Roll a die: $X = \text{number on top face}$

2. Types of Random Variables

A. Discrete Random Variable (DRV)

Definition: A random variable is discrete if it can take a finite or countably infinite number of values.

Properties:

• Takes specific values (integers or counts)
• Probability for each value can be listed
• Probability Sum = 1

Probability Mass Function (PMF):

Example:

• Number of heads in 3 coin tosses → $X = 0,1,2,3$
• Number of defective items in a batch

Graph: Usually bar graph with heights = probabilities.

B. Continuous Random Variable (CRV)

Definition: A random variable is continuous if it can take any value in an interval of real numbers.

Properties:

• Probability of a single exact value = 0
• Probabilities are given over intervals
• Total area under the Probability Density Function (PDF) = 1

Probability Density Function (PDF):

Example:

• Height of students → any value in $140 cm, 200 cm$
• Time required to run 100 m → any value ≥ 0

Graph: Smooth curve representing density.

3. Comparison Table

4. Expected Value and Variance

• Discrete RV:

  $$E(X) = \sum x_i P(X=x_i), \quad Var(X) = \sum (x_i - E(X))^2 P(X=x_i)$$

• Continuous RV:

  $$E(X) = \int_{-\infty}^{\infty} x f(x) dx, \quad Var(X) = \int_{-\infty}^{\infty} (x - E(X))^2 f(x) dx$$

Key Points to Remember

1. Discrete → Countable values, probability for each value.
2. Continuous → Uncountable values, probability over intervals.
3. Always check whether the variable takes specific values or any value in a range.