7. Let M(t)=(1/6)et +(2/6)e²t+(3/6)e³t (Moment generating function of X) a. Find E(x) b. Find Variance (x) nis

1. Let X be an RV with PDF a. Find the mean of X. b. Find the variance of X. f(x) = {¹ -=| - |x| for x < 1 otherwise

Pv. Don't provide handwriting solution

1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.

3. 1 + x Check whether the following function is a valid pdf? If it is valid, obtain its CDF. If not valid, would scaling by a constant make it valid? If so, make it valid. Obtain its CDF? What is its mean? Variance? ƒ = —exp(-√x)U(x)

Let X be a random variable normally distributed with mean μx 1. Find the mean μy of Y. 2. Find the standard deviation oy of Y. 3. Find the PDF fy of Y and evaluate fy (5). (My, oy, fy (5)) = ___________). = 6 and standard deviation ox = 4. Let Y = 4X + 5.

4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.