Continuous Random Variables and Probability Distributions

A set of flashcards covering key concepts in continuous random variables and probability distributions.

What is a continuous random variable?

A continuous random variable is one that takes values in an uncountable set, e.g. height, weight, time.

What is the probability density function (PDF) for a continuous random variable?

A PDF is a function such that f(x) ≥ 0, and the area under the curve within the range equals 1.

How do you determine probabilities from a cumulative distribution function (CDF)?

Use F(X) = P(X ≤ x) = ∫_{-∞}^{x} f(u) du.

What is the formula for the mean (expected value) of a continuous random variable X?

The mean is given by µ = E(X) = ∫_{−∞}^{∞} x f(x) dx.

What is the formula for the variance of a continuous random variable X?

The variance is given by σ² = V(X) = ∫_{−∞}^{∞} (x − µ)² f(x) dx.

What is a cumulative distribution function (CDF)?

The CDF is a function that gives the probability that a random variable X is less than or equal to a certain value x.

What is the relationship between a probability density function (PDF) and a cumulative distribution function (CDF)?

The PDF is the derivative of the CDF, f(x) = dF(x)/dx.

What is the definition of a normal random variable?

A normal random variable has a probability density function defined as f(x) = (1 / (σ√(2π))) e^{−(x − µ)² / (2σ²)}.

What is the empirical rule for normal distribution?

P(µ − σ < X < µ + σ) ≈ 0.6827; P(µ − 2σ < X < µ + 2σ) ≈ 0.9545; P(µ − 3σ < X < µ + 3σ) ≈ 0.9973.

What is a standard normal variable?

A standard normal variable has a mean of 0 and variance of 1, denoted as Z.

Which distributions can be approximated by the normal distribution?

Binomial distribution (if np > 5 and n(1 − p) > 5) and Poisson distribution (if λ ≥ 5).

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

E[h(X)] = ∫_{−∞}^{∞} h(x)f(x) dx.

What is the lack of memory property for exponential distributions?

P(X > t1 + t2 | X > t1) = P(X > t2).

What does the beta distribution probability function look like?

f(x) = (Γ(α + β) / (Γ(α) · Γ(β))) x^(α−1) (1 − x)^(β−1) for 0 < x < 1.

What are the mean and variance formulas for the beta distribution?

Mean: µ = α / (α + β); Variance: σ² = (αβ) / ((α + β)²(α + β + 1)).