lecture 5 - Continuous Random Variables & Probability Distributions

What is a continuous random variable?

A variable that can take any value within a range or interval (e.g. weight, height, time).

How does a continuous variable differ from a discrete one?

Discrete = countable values. Continuous = infinite values within an interval.

What is a probability density function (PDF)?

A function where the area under the curve between two values gives the probability of that range.

What is a cumulative distribution function (CDF)?

Gives the probability that a value is less than or equal to a specific value: P(X ≤ x).

What is the total area under a continuous probability curve?

1 — the entire probability space adds up to 100%.

Why is P(X = exact value) = 0 for continuous variables?

Because there's an infinite number of possible values — we only calculate probability over intervals.

What is the formula for expected value of a continuous variable?

E(X) = ∫ x · f(x) dx (in practice, use known formulas or tables).

What is variance for a continuous variable?

Var(X) = E(X²) - [E(X)]² or computed using integral definitions.

What is a uniform distribution?

A distribution where all outcomes in a range [a, b] are equally likely.

What is the PDF of a uniform distribution?

f(x) = 1 / (b - a) for a ≤ x ≤ b; 0 otherwise.

What is the mean of a uniform distribution?

Mean = (a + b) / 2

What is the variance of a uniform distribution?

Variance = (b - a)² / 12

What is a normal distribution?

A bell-shaped, symmetric distribution defined by mean (μ) and standard deviation (σ).

What is the standard normal distribution?

A normal distribution where μ = 0 and σ = 1.

How do you convert a value to a Z-score?

Z = (X - μ) / σ

What does a Z-score tell us?

How many standard deviations a value is from the mean.

What is the Empirical Rule?

In a normal distribution: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

How do you find P(a < X < b) for normal distributions?

Convert both a and b to Z-scores, then use Z-table to find area between.

If X ~ N(100, 25), what is the Z for X = 115?

Z = (115 - 100) / 5 = 3

What does it mean if Z = -2?

The value is 2 standard deviations below the mean.

What is the area under a normal curve from Z = -∞ to Z = 0?

0.5 or 50% — because it's symmetric around the mean.

When is the normal model appropriate?

When the data is symmetric, unimodal, and bell-shaped; also for large samples (CLT).