STAT 400 Lecture AL1 Examples for 3.1 Spring 2015 Dalpiaz Continuous Random Variables. The probabilities associated with a continuous random variable X are determined by the probability density function of the random variable. The function, denoted (x), must satisfy the following properties: 1. (x) 0 for all x. 2. The total area under the entire curve of (x) is equal to 1.00. Then the probability that X will be between two numbers a and b is equal to the area under (x) between a and b. For any point c, P(X = c) = 0. Therefore, P(a X b) = P(a X b) = P(a X b) = P(a X b). Expected value (mean, average): xxx df )(Xμ. Variance: X2 = dxxx f )(XE2X2X μμ. X2 = 2X222 μ )(XEXE dxxx f. Moment Generating Function: M X ( t ) = E ( e t X ) = - dxxfxte.1. Let X be a continuous random variable with the probability density function f ( x ) = k x , 0 x 4, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? b) Find the cumulative distribution function of X, F X ( x ) = P ( X x ). c) Find the probability P ( 1 X 2 ).d) Find the median of the distribution of X. That is, find m such that P ( X m ) = P ( X m ) = 1/2. e) Find the 30th percentile of the distribution of X. That is, find a such that P ( X a ) = 0.30. f) Find X = E ( X ). g) Find X = SD ( X ).2. Let X be a continuous random variable with the cumulative distribution function F ( x ) = 0, x < 0, F ( x ) = 83 x, 0 x 2, F ( x ) = 1 – 2 1x, x > 2. a) Find the probability density function f ( x ). b) Find the probability P ( 1 X 4 ). c) Find X = E ( X ). d) Find X = SD ( X
View Full Document
Unlocking...