Module 10:Continuous Random Variables and Probability Density Functions

Question 1

what is a continuous random variable?

Answer 1

A varaible whose set of possible values can take on a continuous range. No one is exactly 6’ tall.

Question 2

summarize the properties of a PDF of a continuous random variable

Answer 2

• f(x) > 0
• integrate f(x) dx from negative infinity and infinity equals 1
• If δ is very small, then P(x, x+ δ) roughly equals δ*f(x)
• For an subset B of the real line (B is subset of the reals), P({X is an element of B}) = integral of f(x)dx

Question 3

what is the expectation of a continuous random variable?

Answer 3

Let X be a continuous random variable with probability density function (PDF) f. The expectation (also called the expected value or mean) of X is given by:

E(X) = integral of x*f(x) dx from -infinity to infinity

where we assume absolute convergence of the integral so that the expectation exists. E(X) satisfies the follow properties:
1. For any real valued function g(X), E(g(X)) = integral of g(x) * f(x) dx from negative infinity to infinity.
2. For real valued constants a and b, E(aX +b) = aE(X) + b

Question 4

how do you calc the nth moment of a PDF?

Answer 4

E(X^n) = integral from negative infinity to infinity of x^n * f(x) dx