(i) Let Z = 2X + 3. Then, Cov(Z, Y) = 2Cov(X, Y). (j) If X,...

(i) Let Z = 2X + 3. Then, Cov(Z, Y) = 2Cov(X, Y). (j) If X, Y are uncorrelated, and Z = -5 - Y + X, then Var(Z) = Var(X) + Var(Y). (k) Let X, Y be uncorrelated random variables (Cov(X, Y) = 0) with variance 1 each. Let A = 3X + Y. Then, Var(A) = 4. (l) If X and Y are uncorrelated (meaning that Cov[X, Y] = 0) but are not jointly Gaussian, then E[X|Y] = E[X] always.

Transcript

00:01 So, we are given in the first part of this question, z is equal to 2x plus 3, then convolution of z comma y is equal to 2 convolution of x comma y.

00:23 So, let us get started.

00:25 Means, we have to state whether the statement is true or false.

00:30 See we have given z is equals to 2x plus 3 now if we say the convolution of z comma y so this should become equals to the convolution of 2x plus 3 comma y so that means see 2 times of x for 3 the convolution will be 0 no need to write but for twice x 2 will be taken out and here it should becomes the convolution of x comma y and see the same thing we are getting here that is why the statement first will absolutely become true now let's have a look to second if you see the variance of z is equals to the variance of minus 5 minus y plus x this is given to us.

01:25 So this can be represented like variance of y plus variance of x minus of 2 convolution of x, y.

01:39 So as the convolution of x, y is equal to 0.

01:45 So variance of z would become equal to variance of y plus the variance of x.

01:55 Therefore, the statement j will also become true.

02:06 Now, statement k, v of x is equal to 1, which is also equal to v of y and convolution of x, y is equal to 0.

02:20 So, let us say variance of a would be equal to the variance of 3x plus y.

02:26 Because a where we are considering to be as x that is 3 sorry because a is given to you through x 3x plus y so you have to prove variance of a is equals to 4 so this can be represented as 3 square that means 9 variance of x plus the variance of y plus 2ab so 3 2 is a 6 covariance variance or convolution anything you can say of x y.

03:00 So this will be equals to 9 variance of x is 1 no need to write plus variance of x is 1 plus where covariance of x y is 0.

03:10 So 0 multiplied by 6 will become 0.

03:12 So this will come out equals to 10 not equals to 4...

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