Let X be a continuous random variable with a probability...

Let X be a continuous random variable with a probability density function given by f(x) = {c ? x²; for 0 < x < 1 0; otherwise. Find a) the constant c b) the mean of X c) variance and standard deviation of X d) P(X = .1) e) P(X ? .2) f) P(.1 < X ? .25) j) E(3 X² + 4); that is, the mean of the random variable 3 X² + 4 g) Since we know that V(Y) = E(Y²) ? (E(Y))² for any random variable Y. Find the variance of the random variable Y = 3 X² + 4; that is, V(3 X² + 4).

Transcript

00:01 For this question, the probability dense function is given as, let me just write, so these would be a long solution.

00:09 So this is 0 and 0 x1 and otherwise.

00:14 So the first one, we have to just find the value of c.

00:17 So the probability dance function has the integral value 1.

00:21 So i'm going to just take the integral of this expression and set equal to 0.

00:26 So from here we got c times x cubed over 3 ,0 ,1, which is equal to 1.

00:32 So from here, c over 3, which is equal to 1.

00:38 So from here, c is equal to 3.

00:40 We got the value of c.

00:42 And the mean, so we got a formula for the mean, which is from a to b.

00:47 This is x times f of x, dx.

00:50 So here, a to b, which is the integral from 0 to 1, x times.

00:55 This is 3x squared x x let's take the integral of this expression which is 3 over 4 x to the power 4 0 1 so that would be 3 over 4 this is the mean score for this probable that's the function and in part c the variance so the variance it has the formula which is from a to b so for the variance we have to just find this is x square f of x d x and minus mu squared so we can just write as e of x squared minus e of x barantasse squared so let me just find this one the integral x squared times 3x squared the x minus mu squared so we just got the value of me from the previous case from the part b as 3 over 4 so 3 over 4 squared so if i just take the integral of the first part which is equal to this is three times x to the power 5 over 5 the values the boundary is 0 and 1 and minus 9 over 16 so if i plug in the value which is 3 over 5 minus 9 over 16 that would be 3 over 80 so this is the variance of this expression here and we need to just find the standard deviation so we know that standard deviation squared which is equal to variance so the variance is equal to square of, the standard deviation is equal to square root of variance, so we got 3 over 80.

02:33 If i just put these numbers on the calculator, we got 0 .194.

02:38 So this is a standard deviation for this function here...

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