Bivariate Random Transformation finding CDF

Question

Problem

Assume $Y_i, i=1,2$ are independent with pdf-s $$f(y_i;\theta)=\frac{1}{\theta}e^{\frac{-y_i}{\theta}} \forall y_i>0, \, \theta >0$$

Let $Y = Y_1 + Y_2$, and show that $$F(Y) = P(Y \le y) = 1 - e^{\frac{-y}{\theta}}-\frac{1}{\theta}ye^{\frac{-y}{\theta}}$$

Solution

My teacher solved it by doing this:

\begin{align} P(Y\le y) &= \int_0^y P(Y_1 + Y_2 \le y |Y_2 = u)\frac{1}{\theta}e^{\frac{-u}{\theta}}du \\&= \int_0^y P(Y_1 \le y -u |Y_2 = u)\frac{1}{\theta}e^{\frac{-u}{\theta}}du \\&= \int_0^y (1 - e^{\frac{-(y-u)}{\theta}})\frac{1}{\theta}e^{\frac{-u}{\theta}}du \\&= 1 - e^{\frac{-y}{\theta}}-\frac{1}{\theta}ye^{\frac{-y}{\theta}} \end{align}

My question is why does he use this formula? Do you have any advice on how he arrived at this, and how one could solve this problem by an another method or maybe give me a direction as to what I need to study?

Answer 1

Some explanations of the sequence of derivations\begin{align} P(Y\le y) &\stackrel{\text{(1)}}{=} P(Y_1 + Y_2<y)\\ &\stackrel{\text{(2)}}{=} \mathbb E[\mathbb I_{Y_1 + Y_2<y}]\\ &\stackrel{\text{(3)}}{=} \mathbb E[\mathbb E[\mathbb I_{Y_1 + Y_2<y}|Y_2]]\\ &\stackrel{\text{(4)}}{=} \int_0^{\infty} \mathbb E[\mathbb I_{Y_1 + Y_2<y}|Y_2=u]\,f(u;\theta)\,\text du \\ &\stackrel{\text{(5)}}{=} \int_0^{y} \mathbb E[\mathbb I_{Y_1 + Y_2<y}|Y_2=u]\,f(u;\theta)\,\text du \\ &\stackrel{\text{(2)}}{=} \int_0^{y} P(Y_1 + Y_2 \le y |Y_2 = u)f(u;\theta)\,\text du \\ &= \int_0^{y} P(Y_1 + Y_2 \le y |Y_2 = u)\frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du\\ &\stackrel{\text{(6)}}{=} \int_0^y P(Y_1 \le y -u |Y_2 = u)\frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du \\ &\stackrel{\text{(7)}}{=} \int_0^y P(Y_1 \le y -u)\frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du \\ &\stackrel{\text{(8)}}{=} \int_0^y \left(1 - e^{\frac{-(y-u)}{\theta}}\right)\frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du \\ &= \int_0^y \frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du - \int_0^y e^{\frac{-(y-u)}{\theta}}\,\frac{1}{\theta}e^{\frac{-u}{\theta}}\,\text du \\ &= P(Y_2\le y) - \int_0^y \frac{1}{\theta}e^{\frac{-(y-u+u)}{\theta}}\,\text du\\ &\stackrel{\text{(9)}}{=} 1 - e^{\frac{-y}{\theta}}-\frac{1}{\theta}ye^{\frac{-y}{\theta}}\end{align} where

1. by definition of $Y=Y_1+Y_2$
2. when writing the probability of an event as the expectation of the indicator of this event
3. when using the law of total expectation
4. by definition of the expectation
5. when accounting for the fact that both $Y_1$ and $Y_2$ are positive rv's
6. because $Y_2$ is known to be equal to $y$
7. because $Y_1$ and $Y_2$ are independent $\mathcal E xp(1/\theta)$
8. as there was a mistake of an extra $1/\theta$ in the corresponding row from the question (before my editing), since the cdf of $Y_1$ is $F(y)=1-\exp\{-y/\theta\}$
9. the final result is nonetheless correct.

An alternative approach is to determine first the density of $Y=Y_1+Y_2$ using the convolution theorem: the density of $Y$ writes as \begin{align} f_Y(y) &= \int f_{Y_1}(u)f_{Y_2}(y-u)\,\text du\\ &= \int_0^\infty f_{Y_1}(u)f_{Y_1}(y-u)\,\text du\\ &= \int_0^y \frac{1}{\theta}e^{-u/\theta}\,\frac{1}{\theta}e^{-(y-u)/\theta}\,\text du\,\mathbb I_{y>0}\\ &= \frac{1}{\theta^2} \int_0^y e^{-y/\theta}\,\text du\,\mathbb I_{y>0}\\ &= \frac{ye^{-y/\theta}}{\theta^2}\,\mathbb I_{y>0} \end{align} which is a Gamma$(2,1/\theta)$ density. The probability (cdf) then follows \begin{align} P(Y<y)&=\int_0^y \frac{ue^{-u/\theta}}{\theta^2}\,\text du\\ &=\int_0^y -\frac{u}{\theta}\frac{\text de^{-u/\theta}}{\text du}\,\text du\\ &=-\frac{y}{\theta}e^{-y/\theta}+\int_0^y \frac{\text du}{\text du} \frac{e^{-u/\theta}}{\theta}\,\text du\\ &= -\frac{1}{\theta}ye^{\frac{-y}{\theta}} + 1 - e^{\frac{-y}{\theta}} \end{align}