A probability question regarding two independent uniform distrbutions.

Question

I am thinking about this question: $X_1$ and $X_2$ are independent $Unif(0,1)$ random variables.

(a) Derive the pdf of $\overline{X}=\frac{X_1+X_2}{2}$.

(b) Calculate $E(\frac{X_1}{\overline{X}})$.

(c) Calculate $E(X_1|\overline{X})$, $Var(X_1|\overline{X})$, and $Cov(X_1,\overline{X})$.

First part was just okay. (The triangular-shaped distribution)

My main problem is part (b). I was thinking about either drawing conditional distribution of $X_1|\overline{X}$ or using Basu's theorem somehow... It would be very helpful if you give me a clue for this part. Thank you very much!

Answer 1

By symmetry the integral is the same if you switch $x_1$ for $x_2$ in the numerator, then add.

$$2E = \int_0^1 \int_0^1 2\frac{x_1}{x_1+x_2}dx_1dx_2 + \int_0^1 \int_0^1 2\frac{x_2}{x_1+x_2}dx_1dx_2 = 2$$

So $E = 1$

Answer 2

$$\operatorname{E_{(X_1,X_2)}}\left[\frac{2X_1}{X_1+X_2}\right] = \int_{0}^{1} \int_{0}^{1} \frac{2x_1}{x_1+x_2} \;\mathrm{d}x_2\;\mathrm{d}x_1$$

Then use symmetry