Continuous Random Variables

Discrete vs Continuous

Discrete Random Variable: Countable values (0, 1, 2, ...)
Continuous Random Variable: Uncountable values in interval (all real numbers in range)

Examples of Continuous:

• Height, weight, temperature
• Time (seconds, measured precisely)
• Distance
• Any measurement on continuous scale

Key difference: For continuous variables, P(X = exact value) = 0!

Why P(X = c) = 0?

Infinite possible values in any interval

Probability spread across infinite points → each point has probability 0

Example: Height X uniformly distributed from 60 to 72 inches

• P(X = exactly 65.0000000... inches) = 0
• But P(64 < X < 66) > 0 (interval has positive probability)

Therefore: For continuous variables, focus on intervals, not exact values

Probability Density Function (PDF)

For continuous variable, use PDF: f(x)

Properties:

1. f(x) ≥ 0 for all x
2. Total area under curve = 1
3. P(a < X < b) = area under f(x) from a to b

Key: f(x) is NOT a probability! (Can be > 1)
Area under curve gives probability

Finding Probabilities

P(a < X < b) = area under PDF from a to b

Methods:

• Geometry (for simple shapes: rectangles, triangles)
• Integration (for complex functions)
• Calculator/software (normalcdf for normal distribution)

Example: Uniform distribution on [0, 10]

f(x) = 1/10 for 0 ≤ x ≤ 10

P(3 < X < 7) = (7-3)(1/10) = 4/10 = 0.4

(Rectangle: width 4, height 1/10)

Continuous vs Discrete Probabilities

Discrete: P(X = 5) has meaning

Continuous:

• P(X = 5) = 0
• P(X < 5) = P(X ≤ 5) (no difference!)
• P(3 < X < 7) = P(3 ≤ X ≤ 7) = P(3 < X ≤ 7) = P(3 ≤ X < 7)

All intervals with same endpoints have same probability for continuous variables

Mean of Continuous Random Variable

Mean (Expected Value): $\mu$ or E(X)

For uniform distribution on [a, b]:

$\mu = \frac{a + b}{2}$

Example: X uniform on [0, 10]

$\mu = \frac{0 + 10}{2} = 5$

General: Mean is "balance point" of distribution

Variance and Standard Deviation

For uniform distribution on [a, b]:

$\sigma^2 = \frac{(b-a)^2}{12}$

$\sigma = \frac{b-a}{\sqrt{12}} = \frac{b-a}{2\sqrt{3}}$

Example: X uniform on [0, 10]

$\sigma^2 = \frac{(10-0)^2}{12} = \frac{100}{12} \approx 8.33$

$\sigma = \frac{10}{\sqrt{12}} \approx 2.887$

Uniform Distribution

Simplest continuous distribution

PDF: f(x) = 1/(b-a) for a ≤ x ≤ b

Shape: Flat rectangle (constant height)

Properties:

• Mean: (a+b)/2
• All intervals of same length have same probability
• Symmetric around mean

Example: Bus arrives uniformly between 8:00 and 8:20 (20 minutes)

X = arrival time

• Mean: 10 minutes after 8:00
• P(arrive within first 5 min) = 5/20 = 0.25
• P(arrive between 8:05 and 8:15) = 10/20 = 0.5

Cumulative Distribution Function (CDF)

CDF: F(x) = P(X ≤ x)

For continuous: Integral of PDF from -∞ to x

Properties:

• Increasing function (never decreases)
• lim F(x) as x → -∞ = 0
• lim F(x) as x → ∞ = 1

Use: P(a < X < b) = F(b) - F(a)

Example: Uniform on [0, 10]

$F(x) = \begin{cases} 0 & x < 0 \\ x/10 & 0 \leq x \leq 10 \\ 1 & x > 10 \end{cases}$

P(3 < X < 7) = F(7) - F(3) = 0.7 - 0.3 = 0.4

Normal Distribution (Preview)

Most important continuous distribution (covered in depth in other topic)

Bell-shaped curve

Characterized by: Mean (μ) and standard deviation (σ)

Notation: X ~ N(μ, σ)

Properties:

• Symmetric around mean
• 68-95-99.7 rule
• Use normalcdf on calculator

Percentiles and Quantiles

pth percentile: Value x where P(X ≤ x) = p

Quartiles:

• Q1 (25th percentile): P(X ≤ Q1) = 0.25
• Q2 (50th percentile, median): P(X ≤ Q2) = 0.5
• Q3 (75th percentile): P(X ≤ Q3) = 0.75

Example: Uniform on [0, 10]

Median = 5 (P(X ≤ 5) = 0.5)
Q1 = 2.5 (P(X ≤ 2.5) = 0.25)
Q3 = 7.5 (P(X ≤ 7.5) = 0.75)

Linear Transformations

Same rules as discrete:

If Y = a + bX:

$\mu_Y = a + b\mu_X$

$\sigma_Y = |b|\sigma_X$

Example: Temperature

• C uniform on [0, 40]
• F = 32 + 1.8C
• μ_C = 20, σ_C = 40/√12
• μ_F = 32 + 1.8(20) = 68
• σ_F = 1.8(40/√12) = 72/√12

Combining Independent Variables

Same rules as discrete:

If X and Y independent:

• μ_{X+Y} = μ_X + μ_Y
• σ²_{X+Y} = σ²_X + σ²_Y
• μ_{X-Y} = μ_X - μ_Y
• σ²_{X-Y} = σ²_X + σ²_Y (variances add!)

Common Continuous Distributions

Uniform: Constant PDF on interval
Normal (Gaussian): Bell curve
Exponential: Models time until event (memoryless)
t-distribution: Similar to normal, heavier tails (used in inference)
Chi-square: Used in hypothesis testing

Common Mistakes

❌ Thinking P(X = c) has meaning for continuous X
❌ Interpreting f(x) as probability (it's density, not probability)
❌ Confusing P(X < c) with P(X ≤ c) (they're equal for continuous!)
❌ Adding standard deviations instead of variances
❌ Using discrete formulas for continuous variables

Practice Strategy

1. Identify: Continuous or discrete?
2. Determine distribution: Uniform? Normal? Other?
3. Find parameters: Mean, SD, or a and b for uniform
4. Calculate probability: Use area/geometry or calculator
5. Check: Does answer make sense (between 0 and 1)?

Quick Reference

Continuous: Uncountable values, P(X = c) = 0

PDF: f(x) gives density (not probability)
Area under PDF gives probability

Uniform [a,b]:

• Mean: (a+b)/2
• SD: (b-a)/√12
• P in interval: length/total length

Key: P(a < X < b) = P(a ≤ X ≤ b) for continuous

Remember: For continuous variables, focus on intervals and areas under curves, not individual point probabilities!