
3.5. [Continuous-type random variables II The pdf of a random variable X is given by: L-...
4. (20%) Let X be a continuous random variable with the following PDF Sce-4x 0<x fx(x) = to else where c is a positive constant. (a) (5%) Find c. (b) (5%) Find the CDF of X, Fx(x). (c) (5%) Find Prob{2<x<5} (d)(5%) Find E[X], and Var(X).
Continuous RVs + Conditioned. The PDF of a random variable X is given by: ( ) ?? + 0.4, 0 ≤ ? ≤ 2 0, ??ℎ?????? a. Find the value C that makes fX(x) a valid PDF. (Hint: Draw the PDF to start.) b. Find and sketch the CDF, FX(x). c. Calculate P[X > 1] d. Let A be the event that X > 1. Find and sketch the conditional PDF fX|A(x|A).
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI <x3)
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI
4. Let X and Y be random variables of the continuous type having the joint pdf f(x,y) = 1, 0<x< /2,0 <y sin . (a) Draw a graph that illustrates the domain of this pdf. (b) Find the marginal pdf of X. (c) Find the marginal pdf of Y. (d) Compute plx. (e) Compute My. (f) Compute oz. (g) Compute oz. (h) Compute Cov(X,Y). (i) Compute p. 6) Determine the equation of the least squares regression line and draw it...
(1 point) The pdf of a continuous random variable X is given by 0 otherwise (a) Find E(X). (b) Find Var(X)
Let X and Y be continuous random variables with joint pdf f(x,y) =fX (c(X + Y), 0 < y < x <1 otBerwise a. Find c. b. Find the joint pdf of S = Y and T = XY. c. Find the marginal pdf of T. 、
(b) Let X be a continuous random variable with pdf given by: f(x) =c#x Find the constant c so that f(x) is a pdf of a random variable. C (ii) Find the distribution function F(x)P(X Sx)X (ii) Find the mean and variance of X. .Col니loa, ,iaaa4
Let X be a continuous random variable with the following PDF 6x(1 – x) if 0 < x < 1 fx(x) = 3 0.w. Suppose that we know Y | X = x ~ Geometric(x). Find the posterior density of X given Y = 2, i.e., fxY (2|2).
For a continuous random variable X with the following probability density function (PDF): fX(x) = ( 0.25 if 0 ≤ x ≤ 4, 0 otherwise. (a) Sketch-out the function and confirm it’s a valid PDF. (5 points) (b) Find the CDF of X and sketch it out. (5 points) (c) Find P [ 0.5 < X ≤ 1.5 ]. (5 points)
Consider the following joint PDF of continuous random variables X and Y: 22 – 2pxy + y2 2(1 - 02) where pe(-1,1). (a) Prove that fx,y(x, y) is a joint PDF function. (b) What is the marginal PDF of X? (c) Calculate E[XY] – E[X]E[Y]. (d) Prove that X and Y are independent if and only if p= 0 (e) Show that the conditional PDF of X, given Y = y is N(py, 1 – p2.