B1. Let X be a continuous random variable with PDF 2 f(x) = ½ ½ (2 - x) 9 and f(x) = 0 otherwise. for 1x c (a) Explaining your work, find the value of the constant c. (b) What is P(X > 1)? (c) Calculate the expectation of X. (d) Calculate the variance of X

Let X be a random variable with mean E[X] = 20 and variance Var(X) = 3. Define Y = 3 – 6X. Calculate the mean and variance of Y.

The CDF F(x) of a random variable X is given by;F(x) = { 0          ,           ?? ? ≤ 1              ? (? − 1)^4   ,  ?? 1 < ? ≤ 3                1           ,         ?? ? > 3} find PDF f(x) (ii) ?(|?| < 1.5), (iii) the mean and variance of X.

f(X)= 3/8( 4x - 2x2 )         0<* x <* 2     (* less or equal to) a.  Find the variance of X.

Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.

Show that variance o? = (x²) – ((x))²

Let X and Y be identically distributed with mean 4 and variance 25.(a) If X and Y are uncorrelated, what is the variance of X + Y?(b) If the covariance between X and Y is 8, what is the variance of X + Y? (c) If X and Y are independent, what is E[(X - Y)^2]?

Let X∼Uniform(0,1) distribution. Find the PDF of Y= 3√X and derive the expected value and the variance of Y.

Q6

Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.