Let Y1, Y2, …, Yn be independent random variables, each with...

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ be independent random variables, each with probability density function $$f(y)=\left\{\begin{array}{ll} 3 y^{2}, & 0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ Show that $\bar{Y}$ converges in probability to some constant and find the constant.

Transcript

00:01 So in this question we have a function, f of y, which is 3y squared for y between 0 and 1 and 0 elsewhere, and this is a pdf for a continuous random variable.

00:17 And y1, y2, up to yn are independent and identically distributed according to this function.

00:27 So let's find the expected value of these ys.

00:32 This is the integral from 0 to 1, 3 .3.

00:35 3 y cubed, d .y, where we've inserted an extra factor of y.

00:39 It gives us 3 quarters y to the 4 from 0 to 1, which is 3 quarters.

00:47 The variance of the yis is the expected value of them squared minus their expectation squared.

00:56 So that's the integral from 0 to 1 of 3y to the 4, dy, minus 3 quarters squared.

01:06 So that's 3 5's, y to the 5, from 0 to 1.

01:09 1 minus 9 over 16, which is 3 5 minus 9 over 16, 3 over 5 minus 9 over 16 is 3 over 80.

01:25 So that's the variance of each of the yis.

01:30 Now the central limit theorem tells us that the limit as n goes to infinity of the sum from i equals 1 to n of yi divided by n, so that's the average, minus the mean of the yis divided by the the standard deviation on the mean, so that's the square root of the variance of the y is divided by n, is a standard normal variable.

02:05 So this quantity here is y bar.

02:10 So we have the limit as n goes to infinity of y bar minus three quarters divided by divided by the variance of y i square root is divided by n, so that's root 3 over 80n, is a standard normal variable.

02:40 So what we can say is that if we have z is a standard normal variable, then the limit as n goes to infinity of y bar minus three quarters is equal to the limit as n goes to infinity of the square root of 80n over 3 times z.

03:11 So the probability, so the, what we can say is that the limit as n goes to infinity, that the probability that y bar minus three quarters is in some interval is equal to the limit as n goes to infinity of the probability that 80n over 3, rooted times z, is in that same interval.

03:49 But root 80n over 3 times z is a normal variable.

03:54 It still has mean zero, but we've now multiplied it by this quantity, so it has variance, 80n over 3.

04:09 Sorry, that should be...

04:17 Sorry, we've multiplied by 3 over 80n, so sorry, that should be the other way up...

Question

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ be independent random variables, each with probability density function $$f(y)=\left\{\begin{array}{ll} 3 y^{2}, & 0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ Show that $\bar{Y}$ converges in probability to some constant and find the constant.

Please give Ace some feedback