Suppose that you need to generate a random variable Y with a density function f (y)...
Suppose that you need to generate a random variable Y with a density function f (y) corresponding to a beta distribution with range [0,1], and with a non-integer shape parameter for the beta distribution. For this case there is no closed-form cdf or inverse cdf. Suppose your choices for generating Y are either:
a) an acceptance-rejection strategy with a constant majorizing function g(u) = V over [0, 1], i.e., generate u1 and u2 IID from a U[0,1] generator and accept y = u1 if Vu2<f (u1).
Or,
b) use a numerical approximation to the inverse cdc of Y, say G, generate u from U[0,1] generator, and let y = G(u).
Discuss the advantages and disadvantages of each approach.
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Suppose that you need to generate a random variable Y with a
density function f (y)...
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove...
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possible infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x). 2. Suppose now that X ~ Uniform(0, 1). For each of the distributions listed...
X is a random variable with density function f(x) = x² /3 for -1 < x...
X is a random variable with density function f(x) = x² /3 for -1 < x < 2,0 else. U is uniform(0,1). Find a function g such that g(U) has the same distribution as X.
Suppose that Y has pdf given by f(y)=b/(y^2); y>=b, b>0. Suppose also that U has a...
Suppose that Y has pdf given by f(y)=b/(y^2); y>=b, b>0. Suppose also that U has a uniform(0,1) distribution. What must the function g(*) be so that Z=g(U) has the same distribution as Y?
2. Suppose that you can draw independent samples (U,, U2,U. from uniform distribution on [0,1]. (a) Suggest a method to generate a standard normal random variable using (U, U2,Us...) Justify your...
2. Suppose that you can draw independent samples (U,, U2,U. from uniform distribution on [0,1]. (a) Suggest a method to generate a standard normal random variable using (U, U2,Us...) Justify your answer. b) How can you generate a bivariate standard normal random variable? (Note that a bivariate standard normal distribution is a 2-dimensional normal with zero mean and identity covariance matrix.) (c) What can you suggest if you want to generate correlated normal random variables with covariance matrix Σ= of...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x;...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x; m, 7) = where m is the location parameter, is the scale parameter and they are both constants. We will create a function to simulate draws of a Cauchy(m, 7) distribution in this exercise using the inversion method. (a) (4 points) Derive the cdf, and the inverse function of cdf for Cauchy(m, n). Describe a procedure to generate independent observations from a Cauchy(m, 7)...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F,...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
2. Determine whether the function f(x) is a valid probability distribution (PMF) for a random variable...
2. Determine whether the function f(x) is a valid probability distribution (PMF) for a random variable with the range 0,1,2,3,4 12 f(x) = 30 3. Suppose X is a random variable with probability distribution (PMF) given by f( and a range of 0,1, 2. Find the distribution function (CDF) for X 6
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...
X is a random variable with density function f(x) = x² /3 for -1 < x < 2,0 else. U is uniform(0,1). Find a function g such that g(U) has the same distribution as X.
2. Suppose that you can draw independent samples (U,, U2,U. from uniform distribution on [0,1]. (a) Suggest a method to generate a standard normal random variable using (U, U2,Us...) Justify your answer. b) How can you generate a bivariate standard normal random variable? (Note that a bivariate standard normal distribution is a 2-dimensional normal with zero mean and identity covariance matrix.) (c) What can you suggest if you want to generate correlated normal random variables with covariance matrix Σ= of...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x; m, 7) = where m is the location parameter, is the scale parameter and they are both constants. We will create a function to simulate draws of a Cauchy(m, 7) distribution in this exercise using the inversion method. (a) (4 points) Derive the cdf, and the inverse function of cdf for Cauchy(m, n). Describe a procedure to generate independent observations from a Cauchy(m, 7)...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
2. Determine whether the function f(x) is a valid probability distribution (PMF) for a random variable with the range 0,1,2,3,4 12 f(x) = 30 3. Suppose X is a random variable with probability distribution (PMF) given by f( and a range of 0,1, 2. Find the distribution function (CDF) for X 6
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...
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