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TOPIC:Transformation of random variables.
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Let U U (0,1) and let Y=1-U. Derive an expression for the odf Fy() of Y...
Let U ~uniform(0,1). Let Y =−ln(1−U). hint: If FX (x) = FY (y) and supports x,y...
Let U ~uniform(0,1). Let Y =−ln(1−U). hint: If FX (x) = FY (y) and supports x,y ∈ D, X and Y have the same distribution. Find FY (y) and fY (y). Now, it should be straight forward that Y follows distribution with parameter_____________-
7.1 required non-book problem: Suppose RV Y is continuous with invertible CDF Fy. Then 1. U...
7.1 required non-book problem: Suppose RV Y is continuous with invertible CDF Fy. Then 1. U = Fy (Y) is uniform on the unit interval, i.e., U U (0,1). Recall that this result is known as the Probability Integral Transform. 2. Y = F'(U) has CDF Fy if U U (0,1). Do the following: 1: Let W = Fy(Y) where Fy(y) = 1 - e-dy, osy< where Y is exponential with parameter and Fy is the CDF of Y. Using...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a)...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1)....
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1). Obtain fy and compute E[Y] (b) Now instead assume that Y ~ U(0,1/2) and that X is a continuous random variable. Explain carefully why it is possible to choose fx such that fx (2) = 0 whenever 21 > 1. Obtain an expression linking fx(2) to fx(-x) for 3 € (-1,1). Show that E[X] = -2/3 + 2 S xfx(x) dx. Using your expression...
use R for 4 3. Let U ~ U(0,1), X = cU, and Y = c(1...
use R for 4
3. Let U ~ U(0,1), X = cU, and Y = c(1 -U), where c is a constant. Show that px.y = -1, where p is the correlation. 1. Find estimate of the following integral using simple Monte Carlo method and control variable method Compare two results in terms of sample mean and sample variance.
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y :=...
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by...
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
Let $(x) = 2x2 and let Y = $(X). assume that Y ~ U(0,1/2) and that X is a continuous random variable. fx(x) = 0 whe...
Let $(x) = 2x2 and let Y = $(X). assume that Y ~ U(0,1/2) and that X is a continuous random variable. fx(x) = 0 whenever |2| > 1. Obtain an expression linking fx(x) to fx(-x) for xe (-1,1). Show that E[X] = -2/3 + 28. xfx(x) dx. Using your expression linking fx(x) and fx(-x), obtain an upper bound for E[X] and a pdf fx for which this bound is attained. [10]
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
7.1 required non-book problem: Suppose RV Y is continuous with invertible CDF Fy. Then 1. U = Fy (Y) is uniform on the unit interval, i.e., U U (0,1). Recall that this result is known as the Probability Integral Transform. 2. Y = F'(U) has CDF Fy if U U (0,1). Do the following: 1: Let W = Fy(Y) where Fy(y) = 1 - e-dy, osy< where Y is exponential with parameter and Fy is the CDF of Y. Using...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1). Obtain fy and compute E[Y] (b) Now instead assume that Y ~ U(0,1/2) and that X is a continuous random variable. Explain carefully why it is possible to choose fx such that fx (2) = 0 whenever 21 > 1. Obtain an expression linking fx(2) to fx(-x) for 3 € (-1,1). Show that E[X] = -2/3 + 2 S xfx(x) dx. Using your expression...
use R for 4
3. Let U ~ U(0,1), X = cU, and Y = c(1 -U), where c is a constant. Show that px.y = -1, where p is the correlation. 1. Find estimate of the following integral using simple Monte Carlo method and control variable method Compare two results in terms of sample mean and sample variance.
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
Let $(x) = 2x2 and let Y = $(X). assume that Y ~ U(0,1/2) and that X is a continuous random variable. fx(x) = 0 whenever |2| > 1. Obtain an expression linking fx(x) to fx(-x) for xe (-1,1). Show that E[X] = -2/3 + 28. xfx(x) dx. Using your expression linking fx(x) and fx(-x), obtain an upper bound for E[X] and a pdf fx for which this bound is attained. [10]
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
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