A random variable X has a density function (cx² f(x) = {cx 1< x< 2 2 < x < 3 otherwise Find: a) constant c
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- Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20Let X and Y be independent random variables with density f (x) = 3x² for 0 < x < 1. Then P (X+ Y < 1) is equal to3. Let X be a continuous random variable. Let f(x) = c(x − 1)³ and Sx = [1,3]. Hint: (x - 1)³ = x³ + 3x − 3x² - 1 (a) What value of c will make f(x) a valid density? (b) What is P(X = 2)? (c) Find E(X). (d) What is P(1 < X < 2)?
- Suppose that Y is a continuous random variable. Show EY yfr(y)dy.Suppose that X is a random variable whose density function is defined as follows: (a+1) 2a fx(x) = 2a+1 with 0 -1. For a = 1.15 calculate the IQR of X.For two random variables X and Y, the joint density function is Find (a) the correlation fxy(x, y) = 0.158(x+1) 8(y)+0.18(x) 8(y)+0.18(x)(y-2) 8(x-1)8(y+2)+0.28(x-1)(y-1)+0.05(x-1)(y-3) +0.4
- Suppose that the random variable X has density fx (x) = 4x³ for 0 X).Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.
- b) Let X₁, X₂, Xn, denote a random sample of size n from a distribution with probability density function: f(x;0) = {0(1-x)-(¹+0), x>0 elsewhere Derive fisher's information of 0.Show that if a random variable has a uniform density with parameters α and β, the probability that it will take on a value less than α+p(β-α) is equal to p.1. A continuous random variable X is defined by (3+x) f(x) - 3 sxs- 1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean II