1. Consider two electrical component, A and B, with respective life times X and Y. Assume that a
joint p.d.f. of X and Y is f(x, y) = ??*exp[-(?x+?y)]; x, y>0 and f(x, y)=0 otherwise, where ?
and ? are positive constants. (a) Determine the probability that both components are
functioning at time t. (b) Determine the probability that component A is the first to fail. (c)
Find the distribution Z = X/Y.
Solution
the joint density can be spilt as (pi)*exp[-(Pix] *?*exp[?y)] which suggests that X
and Y are independent exponentials the prob that both the cmponents are wrking at t is
P(x=>t,Y=>t) which we obtain by integration f|(x,y) over the limits x from t to inf and y from t to
inf . b)in this part we need P(Y>X) which can be obtained by integrating the density function
over the limits x from 0 to y and y from 0 to inf. c)Now we observed that X and Y are indep as
we can split their joint density hence X given Y follows exponential only Let Z=X/Y and t?0.
Conditioning on X and applying our characterization to y=X/t, one gets
P(Z?t)=P(Y?X/t)=E(e^?mu*X/t). Now, the density of the distribution of X is pi*e?pi*x on x?0,
hence for every k?0, E(e^?kX)=integral(o to inf)pi*e^?(pi+k)xdx=pi/(pi+k)[?e^?(pi+k)x](with
limits 0 to inf)=pi/(pi+k). Substituting k=mu/t yields the formula.