Suppose X1 and X2 are independent exponential distributions with rate param- eter A and A2, respectively. Show that P{X1 < X2} = x • min{X1, X2} is exponentially distributed with rate parameter A1 + A2.
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- If X has an exponential distribution with the param-eter θ, use the distribution function technique to find the probability density of the random variableY = ln X.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?Suppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.
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- As soon as possible! Let X and Y be continuous random variables with joint PDFSuppose that the lifetime, X, and brightness, Y, of a light bulb are modeled as continuous random variables. Let their joint pdf be given by:f(x,y)=λ_1λ_2e^{-λ_1x-λ_2y},x,y>0 •Are lifetime and brightness independent?•Are lifetime and brightness uncorrelated?For a continuous pdf, the probability of an exact value can be found . Is this true or false ?