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x>0,y>0. Problem 6 Consider the following joint pdf for the random variable X and Y where denotes a unit step function. (a) Find the constant C. (b) Find the marginal PDF's of X and Y. (c) Find the conditional PDF's fx(xY-y) and s, (ylX-x) (d) Find the conditional expected values, EX 1 Y = y} and EX X = Problem 6 Consider the following joint pdf for the random variable X and Y where denotes a unit step function. (a)...
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...
9. Let y= X2, where X has the pdf below (a) Find the mean of Y without finding the pdf of Y. (b) Write the pdf of Y:f<v). (c) Find the mean of Y using.ffy), confirming your answer in part (a).
For each of the following, find the pdf of Y (e) Y =In X and (п+ m+1)! -"(1 - x)" fx (x) = n!m! <1 and m and n are positive integers where 0 (f) Y e and 1 fx(x) e)2/2 where 0 < oo and o2 is a positive constant (i) Y 1 X2 and fx (x)=3(x 1)2, -1 < x <1
Suppose the random variable X has probability density function (pdf) - { -1 < x<1 otherwise C fx (x) C0 : where c is a constant. (a) Show that c = 1/7; (b) Graph fx (х); (c) Given that all of the moments exist, why are all the odd moments of X zero? (d) What is the median of the distribution of X? (e) Find E (X2) and hence var X; (f) Let X1, fx (x) What is the limiting...
Problem 5 20 marks total 0 < y < x2 < Consider two rvs X and Y with joint pdf f(x,y) = k-y, Sketch the region in two dimensions whereAx.y is positive. Then find the constant k and sketch /fx.y) in three dimensions. I4 marksl (a) Find and sketch the marginal pdf/(x), the conditional pdf ffr11/2), and the conditional cdf F11/2) (b) 4 marks] Find P(XcYlY> 1/2), E(XIY=1/2) and E(XlY>1/ 2). /4 marks/ (c) (d) What is the correlation between...
The random variables X1, X2, - .. are independent and identically distributed with common pdf 0 х > fx (x;0) (2) ; х<0. This distribution has many applications in engineering, and is known as the Rayleigh distribution. 2 (a) Show that if X has pdf given by (2), then Y = X2/0 is x2, i.e. T (1, 2) i.e. exponential with mean 2, with pdf fr (y;0) - ; y0; (b) Show that the maximum likelihood estimator of 0 is...
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...
4. Suppose that X and X2 have joint PDF 0 otherwise (a) Use the transformation technique to find the joint PDF of y, and where x,/x, and Y, = X2 (b) Using your answer to part (a), find and identify the distribution of Y.
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is as follows: Px(x)-axexp(-ax) x 20 otherwise where α is a constant. Suppose y = log(x) and y is monotonic in the given range of X. Determine: (i) pdf of y; (ii) valid range of y; and, (iii) expected value of y. Answer hint:J exp(y) (b) Given that, the pdf, namely, fx(x) of a RV, x is uniformly distributed in the range (-t/2, +...