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Probability - Exponential Distribution

  1. Mar 9, 2006 #1
    Here's the evil question:

    Let X~Exponential(alpha). Derive and name the pdf of Y=(alpha)X
     
  2. jcsd
  3. Mar 30, 2006 #2
    Hi! You should show some of your thoughts or working in your post...

    But anyway, here are some steps to guide you along.

    Step 1: Find the cumulative distribution function (cdf) of X. Since X is continuous, you will need to integrate the pdf of X [tex](f(x)=\alpha e^{-\alpha x}, for x\geq 0)[/tex] , with the lower limit being 0 (since we define [tex]x\geq0[/tex] for an exponential distribution) and the upper limit an arbitrary constant x.

    Step 2: Use the relation [tex]Y= \alpha X[/tex] to derive the cdf of Y from the cdf of X. So [tex]F(y) = P(Y\leq y) = P(\alpha X\leq y) = P(X\leq \frac{y}{\alpha})[/tex]

    Step 3: We can calculate this final probability since we know the cdf of X.

    Step 4: Finally, differentiate the cdf of Y to obtain its pdf.

    You will get a nice answer in the end.

    All the best!

    Note: Letters in small casing (e.g. x, y) represent constants while block letters (e.g. X, Y) are used to define the random variables.
     
    Last edited: Mar 30, 2006
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