Example 1b The amount of time,in hours,that a computer functions before breaking down is a continous random var iable with probability density function given by eo0x≥0 f-0 x<0 What is the probabilty that (a)a compupter will funtion between 50 and 150 hours before breaking down; (b)it will funtion less than 100 hours?
Example 1b - /100 , , var 0 () 0 x The amount of time in hours that a computer functions before breaking down is a continous random iable with probability density function given by e x f x λ ≥ = 0 ( ) 50 150 ; ( ) 100 ? x What is the probabilty that a a compupter will funtion between and hours before breaking down b it will funtion less than hours <
Expectation and variance of continuous random variables If X is a continuous random variable having probability density function f(x),then as f(x)dk≈P{x≤X≤x+dx}for dx smal Define the expected value of X by E(X)=xf(x)dx
Expectation and variance of continuous random variables ( ), ( ) { } If X is a continuous random variable having probability density function f x then as f x dx P x X x dx for dx smal ≈ ≤ ≤+ ( ) () . Define the expected value of X by E X xf x dx ∞ −∞ = ∫
Example 2a Find E(X)when the density function of X is m-“8
Example 2a ( ) 2 0 1 () 0 Find E X when the density function of X is x if x f x otherwise ≤ ≤ =
Example 2b The density function of X is given by - f0≤x≤1 otherwise Find E(e)
Example 2b 1 0 1 () 0 ( ). x The density function of X is given by if x f x otherwise Find E e ≤ ≤ =
Proposition 2.1 If X is a continuous random variable with probability density function f(x),then for any real-valued funtion g, ELg(x】=g(x)f(x)dx
Proposition 2.1 ( ), - , [ ( )] ( ) ( ) If X is a continuous random variable with probability density function f x then for any real valued funtion g Egx gx f xdx ∞ −∞ = ∫