Probability - Exponential Distribution

[Brains_Tom]

Here's the evil question:

Let X~Exponential(alpha). Derive and name the pdf of Y=(alpha)X

 

[pizzasky]

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 $$(f(x)=\alpha e^{-\alpha x}, for x\geq 0)$$ , with the lower limit being 0 (since we define $$x\geq0$$ for an exponential distribution) and the upper limit an arbitrary constant x.

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

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.