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Continuous Random Variables 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 F (Y ) P(Y y), y Thus, we have lim F ( y ) 0 and lim F ( y ) 1 y y 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”, P( y Y y y) P( y Y y y) F ( y y) F ( y) F ( y y) F ( y) y F ( y ) dy, as y 0 y • Compare with density in Calculus. mass of a "segment of wire": ( x)dx PDF • For the continuous random variable Y, define the probability density function as d[ F ( y)] f ( y) F ( y) dy for each y for which the derivative exists. Integrating a PDF • Based on the probability density function, we may write y F ( y) f (t )dt Remember the 2nd Fundamental Theorem of Calc.? Properties of a PDF • For a density function f(y): • 1). f(y) > 0 for any value of y. • 2). f (t )dt P(Y ) 1 Problem 4.4 • For what value of k is the following function a density function? ky (1 y ), for 0 y 1 f ( y) otherwise 0, • We must satisfy the property f (t )dt P(Y ) 1 Exponential • For what value of k is the following function a density function? ke0.2 y , for 0 y f ( y) otherwise 0, • Again, we must satisfy the property f (t )dt P(Y ) 1 P(a < Y < b) • To compute the probability of the event a < Y < b ( or equivalently a < Y < b ), we just integrate the PDF: b P(a Y b) F (b) F (a) f (t )dt a 5 F (5) F (3) f (t )dt 3 Problem 4.4 • For the previous density function ky (1 y ), for 0 y 1 f ( y) otherwise 0, • Find the probability P(0.4 Y 1) • Find the probability P(Y 0.4 | Y 0.8) Problem 4.6 • Suppose Y is time to failure and 1 e y , for y 0 F ( y) otherwise 0, 2 • Determine the density function f (y) • Find the probability P(Y 2) • Find the probability P(Y 1 | Y 2)