(a) Suppose X and Y are iid N(0,1) random variables. Show (X+Y)/(√(2)) d= X d= Y . (b) Conversel

(a) Suppose X and Y are iid N(0,1) random variables. Show...

(a) Suppose $X$ and $Y$ are iid $N(0,1)$ random variables. Show $$ \frac{X+Y}{\sqrt{2}} \stackrel{d}{=} X \stackrel{d}{=} Y . $$ (b) Conversely: Suppose $X$ and $Y$ are independent with common distribution function $F(x)$ having mean zero and variance 1 , and suppose further that $$ \frac{X+Y}{\sqrt{2}} \stackrel{d}{=} X \stackrel{d}{=} Y . $$ Show that both $X$ and $Y$ have a $N(0,1)$ distribution. (Use the central limit theorem.)

(a) Suppose $X$ and $Y$ are iid $N(0,1)$ random variables. Show

$$
\frac{X+Y}{\sqrt{2}} \stackrel{d}{=} X \stackrel{d}{=} Y .
$$

(b) Conversely: Suppose $X$ and $Y$ are independent with common distribution function $F(x)$ having mean zero and variance 1 , and suppose further that

$$
\frac{X+Y}{\sqrt{2}} \stackrel{d}{=} X \stackrel{d}{=} Y .
$$

Show that both $X$ and $Y$ have a $N(0,1)$ distribution. (Use the central limit theorem.)

Transcript

00:01 We are given independent random variables x and y, and we're told that both x and y are standard binomial random variables.

00:16 Sorry, standard normal random variables.

00:20 In part a, we're asked to use convolution to show that x plus y is also normal, and to identify its mean and standard deviation.

00:39 Now, using the theorem from this section, we have that, taking w.

00:47 To be x plus y, probability density function w, of outcome w, this is the convolution, the probability density function of x with the probability density function of y.

01:17 By definition, this is the integral from negative infinity to infinity of probability density x at x times the probability density of y.

01:32 At w minus x.

01:35 This is because y is equal to w minus x d x.

01:46 And because we have the both x and y are normally distributed, this is equal to the integral from negative infinity to infinity of 1 over root 2 pi exponential negative x squared over 2 times 1 over root 2 pi.

02:23 Exponential of negative w minus x squared over 2 d x and we see that we can easily factor out 1 over 2 pi so we get 1 over 2 pi integral from negative infinity to infinity and then for the exponent we're going to add the exponents of e together so we get exponential of negative x squared plus w minus squared so like this negative down here all over 2 dx now we're going to complete the square so this is the same as 1 over 2 pi times the integral from negative infinity to infinity of exponential and so to complete the square well we have that this is negative x squared plus well plus w squared minus 2 wx plus x squared over 2 dx.

04:38 Now the x squared plus x squared is 2x squared, so that cancels out with the 2.

04:42 So we have negative x squared, and then we're going to subtract from that w squared minus 2 wx.

04:59 And so we get negative x squared and then plus wx, and then we have negative w squared, and then we have negative w squared.

05:19 And this is going to be w squared over 2.

05:39 And of course this is the same as negative x squared minus wx plus w squared over 2.

05:50 Now to complete the square, we'll want to do divide wx by 2.

06:03 We'll divide negative w by 2, so i have negative w over 2, and we'll square that...

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