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Calculus & Its Applications
- The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.arrow_forwardLet X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.arrow_forwardLet X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.arrow_forward
- Let 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.arrow_forwardSuppose that the random variable X has density ƒx (x) = 4x³ for 0 4X).arrow_forwardLet X₁, X2, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = (0, -, if 0arrow_forwardSuppose that the random variable X has density fx (x) = 4x³ for 0 X).arrow_forwardSuppose that X is a non-negative random variable with continuous density function fx (z) and a > 0. Show that P + P(X> a) ≤ EX E(X)arrow_forward3. 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)?arrow_forwardLet X be a random variable having density function f(x) Sex x ≥ 0 to Find a) E(X), E (X²) and E[(X− 1)²] b) Var (X) and ox otherwisearrow_forwardLet X and Y be continuous random variables with joint probability density function f(x, y) = (3/200y if 0 < 5x < y < 10 O otherwise Find Cov(X, Y)arrow_forwardLet random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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