Variance of sum of random variables with uniform distributions

Question

$X_1$ is random variable with uniform distribution on $(0,1)$. $X_2$ is random variable with uniform distribution on $(0,X_1)$. $X_3$ is random variable with uniform distribution on $(0,X_2)$. Calculate $Var(12(X_1+X_2+X_3))$. I tried to calculate distribution of $X_1+X_2+X_3$, but I have no idea how to do it. Thanks in advance for help.

Answer 1

If you want to callculate $Var(12(X_1+X_2+X_3))$ you do not need calculate the distribution of $X_1+X_2+X_3$.

note if $X\sim Unif(0,b) \Rightarrow E(X^2)=\frac{1}{3} b^2$ $$E(X_1)=\frac{1}{2}$$ $$E(X_2)=EE(X_2|X_1)=E(\frac{X_1}{2})=\frac{1}{4}$$ $$E(X_3)=EE(X_3|X_2)=E(\frac{X_2}{2})=\frac{1}{8}$$

so it is enough to calculate $E(X_1+X_2+X_3)^2$

$$E(X_1+X_2+X_3)^2=E(X_1^2)+E(X_2^2)+E(X_3^2)+2E(X_1X_2)+ 2E(X_1X_3)+2E(X_2X_3)$$

$$E(X_1^2)=\frac{1}{3}$$ $$E(X_2^2)=EE(X_2^2|X_1))=\frac{1}{3}E(X_1^2)$$ $$E(X_3^2)=EE(X_3^2|X_2))=\frac{1}{3}E(X_2^2)$$

$$E(X_1 X_2)=EE(X_1 X_2|X_1)=E(X_1(E(X_2|X_1)))=E(\frac{X_1^2}{2})$$

$$E(X_2 X_3)=EE(X_2 X_3|X_2)=E(X_2(E(X_3|X_2)))=E(\frac{X_2^2}{2})$$

$$E(X_1 X_3)=EE(X_1 X_3|X_1)=E(X_1(E(X_3|X_1)))$$

Just need to calculate $E(X_3|X_1)$

$$E(X_3|X_1)=E(E(X_3|X_1,X_2)|X_1)=E(\frac{X_2}{2}|X_1)=\frac{X_1}{4}$$

Now you have all needs to calculate $Var(12(X_1+X_2+X_3))$