- #1

squenshl

- 479

- 4

## Homework Statement

The random variable ##(x,y)## has density ##f(x,y) = ce^{-(ax+by)}## for ##0\leq y\leq x\leq 1##, with given constants ##a > 0##, ##b > 0##.

1. Compute the constant ##c##.

2. Find the conditional probability density ##f_y(y|x)##.

3. Compute the regression curve of ##Y## on ##X## i.e. ##E(Y|X = x)##.

4. Sketch this regression curve for ##a = 1##, ##b = 2##.

## Homework Equations

## The Attempt at a Solution

For 1. my limits of integration for both ##x## and ##y## is ##0## and ##1##. This gives (after doing the double integration) ##c = \frac{ab}{e^{-(a+b)}-e^{-a}-e^{-b}+1}##. Is this right?

For 2. the conditional density is given by the joint density divided by the marginal density of ##x##. I just want to make sure that my ##c## is correct first.

Not sure on 3. and 4.