Exponential distribution

[Guest]

Hi, thankful if someone can help me out here.

Random variable Y is exponential distributed with expected value 2. Random variable X is equal to Y+(1/2) if 0≤Y≤2 and equals Y-(1/2) if 2<Y≤4. Otherwise X equals 0. What is the expected value of X?

[HallsofIvy]

Pretty straight forward if you know the definitions! The exponential distribution with mean 2 has pdf [tex]\frac{1}{2}e^{-y/2}[/tex]. The expected value of f(y) is [tex]\frac{1}{2}\int_0^\infty f(y)e^{-y/2}dy[/tex]. Since x is y+ 1/2 for [tex]0\le y\le 2[/tex], y- 1/2 for [tex]2< y\le 4[/tex], 0 for all other y, the expected value of [tex]\frac{1}{2}\int_0^2 (y+ 1/2)e^{-y/2}dy+ \int_2^4 (y- 1/2)e^{-y/2}dy[/tex].