Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of X (b) Evaluate the mean and variance ofYifXhas the following PDF (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance of 1 d) Evaluate the mean and variance of Yif X bcos(2RU) where U is a uniform random variable in the unit interval. Problem1 Let Y=aX + b...
Suppose that a rv Y has mgf m(t)- (a) 1-bt) Differentiate this mgf twice and thereby obtain the mean and variance of Y. [5 marksj] (b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t), ie·r(t) = login(1). Find r'() and r"(t), and express r'(0) and r"(0) in terms of EW and VarW. [5 marks] Use the result in (b) to find the mean (d) Find the mean and variance of the...
2) Suppose X is a Normal RV with mean = 17 and variance = 4. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 11 or X > 17) (d) P(X < 11 and X > 17)
Suppose the rv Y is normally distributed with mean -2 and variance 25 (a) Find the probability that Y exceeds 1, given that Y is positive. (b) Find the expected value of Y, given that Y is positive. Hint: For (b), first derive the cdf of the rv given by X = (Y | Y > 0). (Please don't copy and paste)
/20 marks totall Problem 3 Suppose that a rv Y has mgf m(t)- (a) (1- bt)' Differentiate this mgf twice and thereby obtain the mean and variance of Y. 5 marks/ Suppose m(1) is the mgf of a rv W. Let r(t) be the natural logarithm of mo. i.e. r) logm(t). Find rand "(t), and express (0) and "(0) in terms of EW and VarW. (b) 5 marks/ (c) Use the result in (b) to find the mean and variance...
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, +...
Suppose the rv Y is normally distributed with mean -2 and variance 25. Find the expected value of Y, given that Y is positive.
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval. Let Y-ar+b (a) Find the mean...
3.39 PDF of the magnitude of a RV. Find the PDF fy (v) of the RV YX in terms of the PDF of the RV X. Write down the explicit expression for fr () if X~N (0,2). In this case, Y is a Rayleigh distributed RV
1) Suppose X is a Normal RV with mean = 12 and variance = 16. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 16 or X > 12). Hint: Remember to always identify outcomes of interest first! (d) The center of the probability density function of X.