Let X be a continuous random variable with PDF, fx(x) =...

Let X be a continuous random variable with PDF, fx(x) = { ax^2 : 0 < x < 2. 0 : otherwise a) Find a. b) Find variance of X. c) Find Cumulative Distribution Function (CDF) of X. Let X be a continuous random variable with PDF, fx(x) = { ke^{-2x} : x > 0. 0 : otherwise a) Find k. b) Find mean of X. c) Find variance of X. d) Find CDF of X.

Transcript

00:01 So in this question, we're looking at some probability distribution functions.

00:05 So first of all, we have f of x is a x squared for zero less than x less than two, and zero otherwise.

00:18 So first of all, we need to find a.

00:20 So we know that the integral from minus infinity to infinity of fx x d x has got to be one, because there's one probability in terms.

00:31 Total.

00:32 So we get 1 is the integral from 0 to 2 of a x squared the x.

00:39 So this is a third a x cubed from 0 to 2, which is a third times a times 8.

00:50 So a is 3 eighths.

00:54 So that tells us that our probability distribution function is f of x is 3 eighths a x squared.

01:03 If x is in the region 0 to 2, and it's 0 otherwise.

01:19 Okay, let's find the variance of x.

01:23 So the variance of x is the mean of x squared minus the square of the mean of x.

01:39 So this is, the variance of x is the integral x squared f of x d x from 0 to 2, minus the integral from 0 to 2 of x f of x d x squared.

01:57 So this is the integral from 0 to 2 of 3 eighth x to the 4 d x minus 9 over 64 the integral from 0 to 2 of x cubed d x squared so let's do this integral so we get 3 over 4 times 2 to the power of 5 minus 9 over 64 times a quarter times 2 to the power of 4 squared.

02:40 So this is, let's put this in a calculator, 3 over 40 times 2 to the 5 minus 9 over 64 times 1 over 4 times 2 to the power of 4 squared, which gives a variance of 3 over 20 or 0 .15.

03:12 Now we want to find the cumulative distribution function.

03:17 So the cdf of x is the integral from nought to x of f of x dash d x.

03:28 So this is 3 eighths of the integral from nought to x of x squared the x.

03:39 So this is 1 eighth x dash cubed.

03:48 If x is less than, if zero is less than x is less than 2.

03:58 Or, because remember, if we integrate to x greater than 2, then the probability becomes 0.

04:06 So this is just 1 for x greater than 2.

04:10 Or it's 0 for x less than 0.

04:14 Because we integrate from minus infinity up to x.

04:19 So if x is less than zero, we get nothing.

04:22 If x is between 0 and 2, we get this integral.

04:25 And if x is greater than 2, we just get 1.

04:30 Okay, so now let's move on to our second pdf.

04:35 F of x is k e to the minus 2x, if x is greater than 0 or 0 otherwise.

04:46 So first of all, let's find k...

Question

Please give Ace some feedback