Paper 2, Section II, F

Probability | Part IA, 2014

Give the definition of an exponential random variable XX with parameter λ\lambda. Show that XX is memoryless.

Now let X,YX, Y be independent exponential random variables, each with parameter λ\lambda. Find the probability density function of the random variable Z=min(X,Y)Z=\min (X, Y) and the probability P(X>Y)P(X>Y).

Suppose the random variables G1,G2G_{1}, G_{2} are independent and each has probability density function given by

f(y)=C1eyy1/2,y>0, where C=0eyy1/2dyf(y)=C^{-1} e^{-y} y^{-1 / 2}, \quad y>0, \quad \text { where } C=\int_{0}^{\infty} e^{-y} y^{-1 / 2} d y

Find the probability density function of G1+G2G_{1}+G_{2} \cdot [You may use standard results without proof provided they are clearly stated.]

Typos? Please submit corrections to this page on GitHub.