STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 3 1 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 Therefore P X c 0 P a X b P a X b P a X b P a X b X Expected value mean average x f x dx Variance X2 X2 E X X x 2 X 2 f x dx E X E X 2 2 x 2 f x dx X 2 Moment Generating Function MX t E et X e t x f x dx 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 F x 3 x 0 x 0 x 2 8 F x 1 1 x2 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
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