Find the pdf , distribution function of $X$ and $E[(X-2)^2]$

Question

I 'll be very grateful if you can help me , here is the question :

When a person sends an email, the probability that there is an attachment is 0.5. If there is an attachment then the size of the file is exponentially distributed with mean 5 kbytes. Let $X$ denote the size of the attachment received . Find the probability density function, distribution function of $X$ and $E[(X-2)^2]$.

Can you solve this question , i 'll try to understand it by your answers. There are some parts that i don't understand, and i think i can understand them by looking at the answers. Thank you so much.

Answer 1

Denote $Z$ a Bernoulli random variable with $p = \frac12$ and $Y$ an exponential random variable with parameter $\lambda = \frac15$ which represent the size of an attachment if there is one. The two variables are supposed to be independent, then the size of the attachment received is given by $X = ZY$.

Therefore, we have \begin{align*} t\geq 0, \mathbb{P}(X \leq t) &= \mathbb{P}(ZY \leq t) \\ &= \mathbb{P}(ZY \leq t, Z=0) + \mathbb{P}(ZY \leq t, Z=1)\\ &= \frac{1}{2}(1 + \mathbb{P}(Y \leq t))\\ &= \frac{1}{2}(1 + 1 - \exp(-t/5))\\ &= 1 - \frac12\exp(-t/5))\\ \end{align*} In the second equality we used the independance. So the density function is $f(x) = \frac{1}{10}\exp(-x/5)1_{x\geq0}$

Answer 2

Let $A$ be the event that there is an attachment to the email message. Then, $X$ is equal to $Z$ with probability $0.5$ and is zero with probability $0.5$. In other words, $X$ is a mixture of a point mass at zero and a continuous $Z$ with an exponential-like density. Distribution function of $X = 1_{A}Z$ is:

$$ P(X \le x) = P( Z\le x|A)P(A)+ P( X\le x|A^c)P(A^c)= \\ 0.5(1-e^{-\lambda x}) + 0.5 = 1-0.5e^{-\lambda x}, \text{ where } \lambda=1/5. $$

Then, $EX = \int_0^\infty P(X>x)dx= \int_0^\infty 0.5e^{-\lambda x}dx = 0.5/\lambda=2.5$. Note that $X=0$ with probability $0.5$ so that value doesn't contribute to the mean. Note that the continuous part of X has density $0.5 \lambda e^{-\lambda x}$, and hence $VX=0.5/\lambda^2=12.5$.

Furthermore, $E(X-2) = EX-2=0.5$ and $E(X-2)^2 = EX^2-4EX+4= VX + (EX)^2 - 4EX+4 = 12.75.$