Functions of random variable - continuous random variables

[Rupel]

Hi,

I'm can someone please explain the method of obtaining the function of random variables incase of continuous random variables. I'm a bit confused with the procedure given in the combined materials pack.

 

[Bharti Singla]

Suppose we are deriving the PDF of Y , where Y is a function of X. In continuous case, we first need to find CDF. For finding CDF, the steps are:
F(Y)= P(Y<y)
then putting value of Y. Suppose Y= 2x
F(Y)= P(2x<y)
rearranging, F(Y) = P(x<y/2)
Now, it is the prob. of x less than y/2 and we have the distribution of x.
So, F(Y)= integration of (fx)dx to limits 0 to y/2. (say, lower limit of fx is 0)
Now, we have the CDF of Y
Then find PDF of Y by differentiating CDF w.r.t.y
Also, remember to write the ranges of Y. Hop this helps!

 

[Rupel]

Thanks Bharti

 

[Bharti Singla]

You are welcome

 

[Nabil Janmohamed]

Hi,

Can someone help me with the question attached?

Best,

Nabil

 

[vgarg]

Hi,

Can someone help me with the question attached?

Best,

Nabil

Here it is

\begin{align*} \mathbb{E}[X] &= \int_{0}^{\infty} xf_X(x)dx \quad \text{By definition } \\ &= \int_{0}^{\infty} \left ( \int_{0}^{x} \ dt \right )f_X(x)dx \quad \because \int_{0}^{x} \ dt = x \\ &= \int_{0}^{\infty} \left ( \int_{0}^{x}f_X(x) \ dt \right )dx \because f_X(x)\text{ is a constant in the inner integral}\\ &= \int_{0}^{\infty} \int_{0}^{x} f_X(x) \ dt dx\end{align*}

Now note that in the above double integral expression we are integrating first with respect to variable \(t\) holding variable \(x\) constant and then integrating with respect to variable \(x\).



As we see in the above plot we can interchange the order integration above i.e. integrate first wrt variable \(x\) by holding variable \(t\) constant and then integrating wrt variable \(t\) and we get the following

\begin{align*} \mathbb{E}[X] &= \int_{0}^{\infty} \int_{0}^{x} f_X(x) \ dt dx \quad \text{From the above expression} \\ &= \int_{0}^{\infty} \int_{t}^{\infty} f_X(x) \ dx dt \\ &= \int_{0}^{\infty}\left (\int_{t}^{\infty}f_X(x) dx \right ) dt \\ &= \int_{0}^{\infty}\left (1-F_X(t) \right ) dt \end{align*}

Note: This is only valid if rv \(X\) has a support in \(\mathcal{R}_+\)

 

[Nabil Janmohamed]

Thanks, this is immensely helpful.