Before you begin solving probability distribution problems, you need to understand what a Random Variable is and how it maps real-world outcomes into mathematical form. This section transitions you from theoretical probability to real-world modeling through distributions something every UGC NET aspirant must master.
What is a Random Variable?
A random variable is a rule or function that assigns a numerical value to each outcome of a random experiment.
- It does not mean "variable chosen randomly."
- It transforms qualitative outcomes (like coin tosses) into measurable numbers (e.g., 0 = Tail, 1 = Head).
Example
Let’s say you roll a die. The random variable X can represent the number shown on the die:
Sample Space (S): {1, 2, 3, 4, 5, 6}
Random Variable X: X(s) = s
Now, let’s look at the two main types of random variables.
Discrete vs. Continuous Random Variables
a. Discrete Random Variable
This takes on countable values. Think of things you can count: number of students in a class, number of goals in a match, etc.
- Examples: 0, 1, 2, 3...
- Used in situations like binomial or Poisson distributions.
b. Continuous Random Variable
This takes on uncountably infinite values within a range. For example: height, weight, time taken to finish a task.
- Values like 5.3, 7.001, 2.718...
- Used in normal and exponential distributions.
Probability Distribution:
A Probability Distribution describes how the probabilities are distributed over the values of the random variable.
In simple terms, it answers: “What’s the chance of each possible value?”
It can take different forms depending on the type of random variable.
For Discrete Random Variables:
You use a Probability Mass Function (PMF)
For Continuous Random Variables:
You use a Probability Density Function (PDF)
Probability Mass Function (PMF)
The PMF gives the probability that a discrete random variable is exactly equal to some value.
Formula:
P(X = x) = p(x), where 0 ≤ p(x) ≤ 1 and Σp(x) = 1
Example:
Let X = Number of heads in 2 coin tosses.
- Possible values of X: {0, 1, 2}
- P(X = 0) = 1/4
- P(X = 1) = 1/2
- P(X = 2) = 1/4
Check: 1/4 + 1/2 + 1/4 = 1
Probability Density Function (PDF)
The PDF is used with continuous random variables. It does not give probabilities directly. Instead, it gives a density. You calculate the probability using an integral.
Key Points:
- For any continuous variable, P(X = a) = 0 (point probability is zero).
- Instead, we calculate P(a ≤ X ≤ b) using area under the curve.
Properties:
f(x) ≥ 0 for all x ∫ f(x) dx over all x = 1
Think of PDF as a "shape" describing how likely different values are in a continuous range.
Cumulative Distribution Function (CDF)
The CDF of a random variable gives the probability that the variable takes on a value less than or equal to a specific number. It works for both discrete and continuous variables.
Formula (Discrete):
F(x) = P(X ≤ x) = Σ P(X = k), where k ≤ x
Formula (Continuous):
F(x) = ∫-∞x f(t) dt
Example (Discrete):
Let X = number of heads in 2 coin tosses. We already have:
- P(X = 0) = 1/4
- P(X = 1) = 1/2
- P(X = 2) = 1/4
Then, the CDF values:
- F(0) = P(X ≤ 0) = 1/4
- F(1) = P(X ≤ 1) = 1/4 + 1/2 = 3/4
- F(2) = P(X ≤ 2) = 1 (as it includes all outcomes)
Expected Value (Mean) of a Random Variable
The expected value is the long-run average value of repetitions of the experiment it represents. In other words: “What do we expect to happen on average?”
Formula for Discrete:
E(X) = Σ x × P(x)
Formula for Continuous:
E(X) = ∫ x × f(x) dx
Example (Discrete):
Suppose X = {0, 1, 2} with probabilities: P(0) = 0.2, P(1) = 0.5, P(2) = 0.3
E(X) = 0×0.2 + 1×0.5 + 2×0.3 = 0 + 0.5 + 0.6 = 1.1
Variance and Standard Deviation of Random Variables
Variance measures the spread or variability of a random variable. A smaller variance means values are closer to the mean.
Formula for Variance (Discrete):
Var(X) = E[(X - E(X))²] = Σ (x - μ)² × P(x)
Alternative Shortcut Formula:
Var(X) = E(X²) - [E(X)]²
Standard Deviation:
σ = √Var(X)
Example:
Continuing the previous example, let’s find variance.
- E(X) = 1.1
- E(X²) = 0²×0.2 + 1²×0.5 + 2²×0.3 = 0 + 0.5 + 1.2 = 1.7
- Var(X) = 1.7 - (1.1)² = 1.7 - 1.21 = 0.49
- Standard Deviation = √0.49 = 0.7
Summary
| Concept | Discrete | Continuous |
|---|---|---|
| Function Type | PMF | |
| Point Probability | P(X = x) | P(X = x) = 0 |
| Probability Over Interval | Sum of values | Area under curve (integral) |
| CDF Usage | Σ P(X ≤ x) | ∫ f(t) dt |
Real-World Applications
- Insurance Industry: Models expected claims using continuous distributions (e.g., exponential or gamma).
- Banking & Risk: Uses normal and binomial distributions to estimate risk of default or fraud.
- Marketing: Predicts customer lifetime value using expected value of customer behavior.
- Operations Research: Queue theory depends heavily on Poisson and exponential distributions.
Final Reflection
Ever wondered why statistics is both a science and an art? It's because tools like random variables and distributions allow us to simplify chaos without losing accuracy. They teach us to quantify uncertainty.
Next time you see a pattern in stock prices or sales forecasts—remember: there's a random variable behind it, silently shaping the outcome.
Stay curious. Practice a few PMF/PDF problems tonight. The real world is waiting to be decoded.