The probability density function of random variable X is given as f x ( x ) = λ e − λ x , x ≥ 0. A new random variable Y = e − λ X is formed. Find the PDF of Y.

Question

The probability density function of random variable X is given as       f    x    (  x  )  =  λ      e          −      λ      x        ,  x  ≥  0. A new random variable   Y  =      e          −      λ      X       is formed. Find the PDF of Y.

BergweiseiEK · Accepted Answer

There are shortcuts, but we will use a basic method. The idea is to find the cumulative distribution function of Y, and then differentiate to find the density function. We have      F    Y    (  y  )  =  Pr  (  Y  ≤  y  )  =  Pr  (      e          −      λ      X        ≤  y  )  =  Pr  (  −  λ  X  ≤  log  ⁡  y  )  .Thus      F    Y    (  y  )  =  Pr      (    X    ≥    −                  log        ⁡        y            λ        )    .So we know that   Pr  (  X  ≥  t  )  =      e          −      λ      t      , if t is positive. If t is negative, the probability is 1.Substitute for t. There is dramatic simplification. The   λ&#039;s cancel, and we get       e          log      ⁡      y      , that is y. But note this is correct only when   −  log  ⁡  y is   ≥  0, that is, when   y  ≤  1. If       F    Y    (  y  )  =  1.

The probability density function of random variable X is given as f_x(x)=lambda e^(-lambda x), x>=0. A new random variable Y=e^(-lambda X) is formed. Find the PDF of Y.

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