Random Variables | StatsMasters

What is a Random Variable?

A random variable is a numerical outcome of a random process. It assigns a number to each possible outcome in a sample space.

Random Variables

Experiment: Roll a die

Random variables:

X = the number showing (1, 2, 3, 4, 5, or 6)
Y = 1 if even, 0 if odd
Z = X² (the square of the number)

Types of Random Variables

Discrete Random Variables

Takes on countable values (often integers).

Discrete Examples

Number of heads in 10 coin flips (0, 1, 2, …, 10)
Number of customers per hour (0, 1, 2, …)
Number of defects in a batch (0, 1, 2, …)
Number of siblings (0, 1, 2, 3, …)

Continuous Random Variables

Takes on any value in an interval (uncountably infinite).

Continuous Examples

Height of a person (any value in a range)
Time until an event occurs
Temperature readings
Weight, distance, duration

Probability Mass Function (PMF)

For discrete random variables, the PMF gives the probability of each value.

PMF Definition

$P(X = x) = f(x)$

Properties:

$f(x) \geq 0$ for all x
$\sum_{\text{all } x} f(x) = 1$

PMF: Roll a Die

X = number on a fair die

x	P(X = x)
1	1/6
2	1/6
3	1/6
4	1/6
5	1/6
6	1/6

Sum: 1/6 × 6 = 1 ✓

PMF: Number of Heads in 2 Flips

X = number of heads when flipping 2 coins

Possible outcomes: HH, HT, TH, TT

x	Outcomes	P(X = x)
0	TT	1/4
1	HT, TH	2/4 = 1/2
2	HH	1/4

Sum: 1/4 + 1/2 + 1/4 = 1 ✓

Probability Density Function (PDF)

For continuous random variables, probabilities are defined for intervals, not individual points.

PDF Definition

$P(a \leq X \leq b) = \int_a^b f(x) \, dx$

Properties:

$f(x) \geq 0$ for all x
$\int_{-\infty}^{\infty} f(x) \, dx = 1$
$P(X = x) = 0$ for any single point

PDF Example

X is uniform on [0, 2], meaning f(x) = 0.5 for 0 ≤ x ≤ 2.

What’s P(0.5 ≤ X ≤ 1.5)?

Area under curve from 0.5 to 1.5: = 0.5 × (1.5 - 0.5) = 0.5 × 1 = 0.5

Cumulative Distribution Function (CDF)

The CDF gives the probability that X is less than or equal to some value.

CDF Definition

$F(x) = P(X \leq x)$

For discrete: $F(x) = \sum_{k \leq x} P(X = k)$

For continuous: $F(x) = \int_{-\infty}^{x} f(t) \, dt$

CDF Properties

F(x) is non-decreasing
$\lim_{x \to -\infty} F(x) = 0$
$\lim_{x \to \infty} F(x) = 1$
$P(a < X \leq b) = F(b) - F(a)$

CDF: Die Roll

X = number on a die

x	P(X = x)	F(x) = P(X ≤ x)
1	1/6	1/6
2	1/6	2/6
3	1/6	3/6 = 1/2
4	1/6	4/6
5	1/6	5/6
6	1/6	6/6 = 1

P(X ≤ 4) = F(4) = 4/6 = 2/3

Expected Value (Mean)

The expected value is the long-run average value of the random variable.

Expected Value

Discrete: $E(X) = \mu = \sum_{x} x \cdot P(X = x)$

Continuous: $E(X) = \mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$

Expected Value: Die Roll

X = number on a fair die

E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6)

E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 3.5

Note: You can never roll exactly 3.5, but over many rolls, the average approaches 3.5.

Properties of Expected Value

Expected Value Properties

$E(c) = c$ (constant)
$E(cX) = c \cdot E(X)$
$E(X + Y) = E(X) + E(Y)$ (always true!)
$E(X - Y) = E(X) - E(Y)$
$E(aX + b) = a \cdot E(X) + b$

Variance

Variance measures how spread out the distribution is around the mean.

Variance

$Var(X) = \sigma^2 = E[(X - \mu)^2] = E(X^2) - [E(X)]^2$

Standard deviation: $\sigma = \sqrt{Var(X)}$

Variance: Die Roll

We know E(X) = 3.5

First find E(X²): E(X²) = 1²(1/6) + 2²(1/6) + 3²(1/6) + 4²(1/6) + 5²(1/6) + 6²(1/6) E(X²) = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6 ≈ 15.17

Var(X) = E(X²) - [E(X)]² Var(X) = 91/6 - (3.5)² = 91/6 - 12.25 ≈ 2.92

SD(X) = √2.92 ≈ 1.71

Properties of Variance

Variance Properties

$Var(c) = 0$ (constant has no variance)
$Var(cX) = c^2 \cdot Var(X)$
$Var(X + c) = Var(X)$ (shifting doesn’t change spread)
$Var(aX + b) = a^2 \cdot Var(X)$

For independent X and Y: 5. $Var(X + Y) = Var(X) + Var(Y)$ 6. $Var(X - Y) = Var(X) + Var(Y)$ (adds, not subtracts!)

Summary Table

Concept	Discrete	Continuous
Probability function	PMF: P(X = x)	PDF: f(x)
P(single value)	Can be non-zero	Always 0
Sum/Integral	∑ P(X = x) = 1	∫ f(x)dx = 1
E(X)	∑ x·P(X = x)	∫ x·f(x)dx
Var(X)	E(X²) - [E(X)]²	E(X²) - [E(X)]²

Summary

In this lesson, you learned:

Random variables assign numbers to random outcomes
Discrete RVs have countable values; continuous RVs have any value in an interval
PMF gives probabilities for discrete RVs; PDF for continuous
CDF F(x) = P(X ≤ x) for both types
Expected value E(X) is the long-run average
Variance Var(X) measures spread around the mean
E(X + Y) = E(X) + E(Y) always
Var(X + Y) = Var(X) + Var(Y) for independent RVs

Practice Problems

1. X has PMF: P(X = 1) = 0.3, P(X = 2) = 0.5, P(X = 3) = 0.2 Find E(X).

2. For the same X, find Var(X).

3. If E(X) = 10 and Var(X) = 4, find E(3X + 5) and Var(3X + 5).

4. X is uniform on [0, 10]. What’s P(2 ≤ X ≤ 7)?

5. Why is P(X = 5.0000) = 0 for a continuous RV, even if 5 is a possible value?

Click to see answers

1. E(X) = 1(0.3) + 2(0.5) + 3(0.2) = 0.3 + 1.0 + 0.6 = 1.9

2. E(X²) = 1²(0.3) + 2²(0.5) + 3²(0.2) = 0.3 + 2.0 + 1.8 = 4.1

Var(X) = E(X²) - [E(X)]² = 4.1 - 1.9² = 4.1 - 3.61 = 0.49

3. E(3X + 5) = 3·E(X) + 5 = 3(10) + 5 = 35 Var(3X + 5) = 3²·Var(X) = 9(4) = 36

4. For uniform [0, 10], f(x) = 1/10 P(2 ≤ X ≤ 7) = (7-2)/10 = 0.5

5. For continuous RVs, probability equals area under the PDF. A single point has width 0, so area = 0. Probabilities only exist for intervals, not points.

Next Steps

Now explore specific distributions:

Binomial Distribution - Discrete success/failure trials
Normal Distribution - The bell curve
Discrete Distributions - Binomial, Poisson, more

What is a Random Variable?

Types of Random Variables

Discrete Random Variables

Continuous Random Variables

Probability Mass Function (PMF)

Probability Density Function (PDF)

Cumulative Distribution Function (CDF)

CDF Properties

Expected Value (Mean)

Properties of Expected Value

Variance

Properties of Variance

Summary Table

Summary

Practice Problems

Next Steps

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