Question

1.

a. Show that for any y ∈ Rⁿ, show that yy^T is positive semidefinite.

b. Let X be a random vector in Rⁿ with covariance matrix Σ = E[(X − E[X])(X − E[X])^T]. Show that Σ is positive semidefinite.

2. Let X and Y be real independent random variables with PDFs given by f and g, respectively. Let h be the PDF of the random variable Z = X + Y .

a. Derive a general expression for h in terms of f and g

b. If X and Y are both independent and uniformly distributed on [0, 1] (i.e. f(x) = g(x) = 1 for x ∈ [0, 1] and 0 otherwise) what is h, the PDF of Z = X + Y ?

Please show your work. Thanks!

Answer 1

1 a) A matrix is positive semidefinite if for all we have . Now, let . Then for any we have

because . Hence, is positive semidefinite.

b) Fix an ; let be the constant random vector , defined on the same probability space as . Then and are independent (because is constant). Therefore (alternatively, by linearity of expectation), we have

Note that and are scalars (and are transpose of each other); therefore, they are equal, which means

Thus, we have

for all . Hence, is positive semidefinite.

2 a) We have by independence of the random variables

where

Thus,

b) In this case, we have

If then , which means

if .

Let ; then either , or , or . In the first case, we have

in the second case

In the third case,

Therefore, the PDF is