- #1

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Here's the evil question:

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

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

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- Thread starter Brains_Tom
- Start date

- #1

- 6

- 0

Here's the evil question:

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

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

- #2

- 172

- 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.

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.

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