Let Z1 and Z2 be independent standard normal rvs and let W1=Z1 W2=ρ·Z1+√(1-ρ^2) Z2 (a) By defi

Let Z1 and Z2 be independent standard normal rvs and let...

Let $Z_{1}$ and $Z_{2}$ be independent standard normal rvs and let $W_{1}=Z_{1} \quad W_{2}=\rho \cdot Z_{1}+\sqrt{1-\rho^{2}} Z_{2}$ (a) By definition, $W_{1}$ has mean 0 and standard deviation $1 .$ Show that the same is true for $W$ (b) Use the properties of covariance to show that Cov $\left(W_{1}, W_{2}\right)=\rho$ . (c) Show that $\operatorname{Corr}\left(W_{1}, W_{2}\right)=\rho$

Let $Z_{1}$ and $Z_{2}$ be independent standard normal rvs and let
$W_{1}=Z_{1} \quad W_{2}=\rho \cdot Z_{1}+\sqrt{1-\rho^{2}} Z_{2}$
(a) By definition, $W_{1}$ has mean 0 and standard deviation $1 .$ Show that the same is true for $W$
(b) Use the properties of covariance to show that Cov $\left(W_{1}, W_{2}\right)=\rho$ .
(c) Show that $\operatorname{Corr}\left(W_{1}, W_{2}\right)=\rho$

Key Concepts

Standard Normal Distribution

The standard normal distribution is a probability distribution with mean zero and unit variance. It is a special case of the normal distribution and is central to many statistical techniques, especially because the behavior and properties of normal random variables are well understood and can be easily manipulated in linear transformations.

Linear Transformation of Random Variables

A linear transformation of a random variable involves scaling and/or shifting the variable. When applied to normally distributed variables, the resulting variable remains normally distributed. In the problem, W? is defined as a linear combination of independent standard normal random variables, and its mean and variance can be computed using the linearity of expectation and the properties of variance for such combinations.

Covariance

Covariance measures the degree to which two random variables change together. It is defined as the expected product of the deviations of the two variables from their respective means. Properties of covariance, such as bilinearity and the fact that the covariance between independent random variables is zero, are essential in determining the relationship between variables in linear combinations.

Correlation

Correlation is a standardized measure of the relationship between two variables, obtained by dividing their covariance by the product of their standard deviations. It yields a dimensionless quantity that lies between -1 and 1, indicating both the strength and direction of a linear relationship. In many problems involving linear transformations of normally distributed variables, once the covariance and variances are known, computing the correlation becomes straightforward.

Transcript

00:02 We are given independent standard normal random variables, z1 and z2, and we're given w1 is z1, and that w2 is row times z1 plus square root of 1 minus row squared z2, or row is the correlation coefficient of z1 and z2.

00:28 In part a, we're asked to show that the standard deviation of w2 is 1 ,000, and 2, is 1.

00:36 And w2 has a mean of 0? well, first of all, we have the expected value of w2.

00:51 By linearity, this is going to be, by definition, expected value of row z1 plus square root of 1 minus row squared times z2, and this is equal to, since row is a constant, row times the expected value of z1 plus the square root of 1 minus row squared times the expected value of z2.

01:23 Now we have that the expected values of z1 and z2 are both 0.

01:31 So this becomes row times 0 plus the square root of 1 minus row squared times 0 because these are normal random variables.

01:46 This is simply 0.

01:48 So we've shown that the mean is 0.

01:53 And moreover, we have that the variance of w2.

02:02 Again, this is the variance by definition of row times z1 plus the square root of 1 minus row squared times z2.

02:13 And because z1 and z2 are independent, this is going to be row squared times the variance of z1 plus square root of 1 minus row squared, which is simply 1 minus row squared, times the variance of z2, and this is because z1 and z2 are independent, so their covariance term cancels out.

02:53 I guess row is not really their covariance here.

02:56 It's something else...

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