5. (Discrete and ontinuous random variables) (a) Consider a CDF of a random variable X, 10...
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
3. (10 points) Let X be a continuous random variable with CDF for x < -1 Fx(x) = { } (x3 +1) for -1<x<1 for x > 1 and let Y = X5 a. (4 points) Find the CDF of Y. b. (3 points) Find the PDF of Y. c. (3 points) Find E[Y]
(a) Below is the CDF for a discrete random variable, X if x 1 1/2 if 1 x< 2 if 2 x 3 7/8 if 3 x 4 F(x) = 3/4 2 1 if nx <n+1. Describe the probability 2n In general, note that for any positive integer n, F(x) distribution of X by finding P(X 1), P(X = 2), P(X positive integer n, and describe an experiment that would result in this random variable X. 3), and the general...
Define the random variable Y = -2X. Determine the cumulative
distribution function (CDF) of Y . Make sure to completely specify
this function. Explain.
Consider a random variable X with the following probability density function (PDF): s 2+2 if –2 < x < 2, fx(x) = { 0 otherwise. This random variable X is used in parts a, b, and c of this problem.
Problem # 8. a) Let X be a continuous random variable with known CDF FX(x). LetY = g(X) where g(·) is the so-called signum function, which extracts the sign of its argument. In other words, g(X) = { -1 x<0, 0 x=0, 1 x>0 } Express the PDF fY (y) in terms of the known CDF FX(x). b) Let X be a random variable with PDF: fX(x) = { x/2 0 <= x < 2, 0 otherwise} Let Y be...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
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Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) 0.5. Find the median for random variables with the following density functions (a) f(x) = e-*, x 0 (b) f(x) = 1, 0 〈 x 1. (c) f(x) 6x(1 - x),0 <1.
6. Consider a random variable X with pdf given by 0 elsewhere. (a) What is c? Plot the pdf. (b) Plot the cdf of X. Find P(X 0.5< 0.3).
A continuous random variable X has cdf F given by: F(x)x3, x e [0,1] (1, x〉1 a) Determine the pdf of X b) Calculate Pi<X <3/4 c) Calculate E X]
2). Consider a discrete random variable X whose cumulative distribution function (CDF) is given by 0 if x < 0 0.2 if 0 < x < 1 Ex(x) = {0.5 if 1 < x < 2 0.9 if 2 < x <3 11 if x > 3 a)Give the probability mass function of X, explicitly. b) Compute P(2 < X < 3). c) Compute P(x > 2). d) Compute P(X21|XS 2).