econ 120A Topic 4

Continuous Random Variable (RV)

Can take on any value in an interval on the real line or in a collection of intervals. We cannot list all the possible values of a continuous random variable.

Probability of a Single Value

The probability of a continuous random variable assuming any particular value is always zero. Therefore, we do not discuss the probability of a continuous random variable assuming a particular value.

Probability Density Function (PDF), f(x)

Represents the probability distribution of a continuous variable X. It is used to find the probability of a random variable assuming a value within a given interval of real numbers. Every continuous random variable X has a PDF f(x).

Area Under the PDF

The total area under the curve (the PDF) is always 1. The area under the PDF f(x) between a and b tells us the probability that the value of X falls within the interval (a,b).

Calculating Interval Probability

The probability of X assuming a value between c and d, P(c≤X≤d), is the area under the PDF between c and d. This is calculated by integrating the PDF: P(a<X<b)=∫ₐᵇ f(x)dx.

Interval Probability Equivalence

For continuous random variables, because the probability of X equaling a specific value is zero, the inclusion of endpoints does not change the probability: P(c≤X≤d)=P(c<X<d).

Uniform Probability Distribution

A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. It is also called the rectangular probability distribution.

Uniform PDF Formula

The uniform PDF is f(x)=1/(b−a) for a≤x≤b, and 0 elsewhere.

Uniform Distribution Mean (μ)

The mean of X is E(X)=μ=(a+b)/2.

Uniform Distribution Variance (σ²)

The variance is Var(X)=σ²=(b−a)²/12.

Normal Probability Distribution

The most common continuous distribution; used for heights, rainfall, test scores, etc. Also called “Gaussian.”

Normal Distribution Notation

X∼N(μ,σ²) means X is normally distributed with mean μ and variance σ².

Normal Curve Properties

Peak at the mean (also median and mode); symmetric; tails extend to infinity.

Area Property of Normal Distribution

Total area is 1; 0.50 left of mean, 0.50 right.

Standard Normal Distribution

A normal distribution with μ=0 and σ=1. Denoted Z∼N(0,1).

Z-Transformation (Standardization)

Z=(X−μ)/σ converts X to a standard normal variable.

Calculating P(Z<a)

The area to the left of a, given by the Z-table

Calculating P(a<Z<b)

P(a<Z<b)=P(Z<b)−P(Z<a). Use Z-table values.

Calculating P(Z>a)

P(Z>a)=1−P(Z<a) or equivalently P(Z<−a).