Continuous Random Variables

For discrete random variables, we required that Y was limited to a finite (or countably infinite) set of values. Now, for continuous random variables, we allow Y to take on any value in some interval of real numbers. As a result, P(Y = y) = 0 for any given value y.

CDF For continuous random variables, define the cumulative distribution function F(y) such that Thus, we have

A Non-Decreasing Function Continuous random variable Continuous distribution function implies continuous random variable.

“Y nearly y” P(Y = y) = 0 for any y. Instead, we consider the probability Y takes a value “close to y”, Compare with density in Calculus.

PDF For the continuous random variable Y, define the probability density function as for each y for which the derivative exists.

Integrating a PDF Based on the probability density function, we may write Remember the 2 nd Fundamental Theorem of Calc.?

Properties of a PDF For a density function f(y): 1). f(y) > 0 for any value of y. 2).

Problem 4.4 For what value of k is the following function a density function? We must satisfy the property

Exponential For what value of k is the following function a density function? Again, we must satisfy the property

P(a < Y < b) To compute the probability of the event a < Y < b ( or equivalently a < Y < b ), we just integrate the PDF:

Problem 4.4 For the previous density function Find the probability

Problem 4.6 Suppose Y is time to failure and Find the probability Determine the density function f (y)