Chapter 5. Continuous Random Variables

Continuous Random Variables Discrete random variables –Random variables whose set of possible values is either finite or countably infinite. Continuous random variables –Random variables whose set of possible values is uncountable. –The lifetime of a light bulb. –The amount of precipitation in a year. 2

Definition of Continuous Random Variable Probability of continuous random variable –If there exists a nonnegative function f, defined for all real x (-∞, ∞), having the property that for any set B of real numbers The function f is called the probability density function of the random variable X. 3

The probability of the whole sample space is 1. The probability of a certain region B = [a,b] is The probability at a certain point is zero 4

Ex 1a. Suppose that X is a continuous random variable whose probability density function is given by (a) What is the value of C? (b) Find P(X > 1) 5

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Ex 1b. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given by What is the probability that (a) a computer will function between 50 and 150 hours before breaking down; (b) it will function less than 100 hours? 7

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Ex 1c. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by What is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events E i, i = 1, 2, 3, 4, 5, that the i-th such tube will have to be replaced within this time, are independent. 9

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Distribution Function and PDF Probability density function can be considered as the measure of how likely that the random variable will be near a. 11

Ex 1d. If X is continuous with distribution function F X and density function f x, find the density function of Y = 2X. 12

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Expectation and Variance of Continuous Random Variables Discrete random variable Continuous random variable 14

Ex 2a. Find E[X] when the density function of X is 15

Ex 2b. The density function of X is given by Find E[e X ] 16

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Proposition 2.1 If X is a continuous random variable with probability density function f(x), then for any real-valued function g, 18

Lemma 2.1 For a nonnegative random variable Y, 19

Proof of proposition

Ex 2c. A stick of length 1 is split at a point U that is uniformly distributed over (0,1). Determine the expected length of the piece that contains the point p, 0 ≤ p ≤ 1. 21

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Corollary 2.1 If a and b are constants, then E[aX + b] = aE[X] + b Variance of continuous random variable Var(X) = E[(X-μ) 2 ] Var(X) = E[X 2 ] - (E[X]) 2 23

Ex 2e. Find Var(X) if X has the pdf 24