Homework Answers
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find...
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
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. Let X be a continuous r.v. with pdf fx() for which EX exists. We want to find the real number ...
2. Let X be a continuous r.v. with pdf fx() for which EX exists. We want to find the real number a that minimizes EX-al over all real a. Determine an equation involving the CDF Fx(-) that can be solved to obtain a*.
2. Let X be a continuous r.v. with pdf fx() for which EX exists. We want to find the real number a that minimizes EX-al over all real a. Determine an equation involving the CDF Fx(-) that...
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...
Let X, Y be iid random variables that are both uniformly distributed over the interval (0,1)....
Let X, Y be iid random variables that are both uniformly distributed over the interval (0,1). Let U = X/Y. Calculate both the CDF and the pdf of U, and draw graphs of both functions.
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X). b) Show that Z=-2ln(Y) has a Gamma dist. & derive it. 4. X_i ~ cont with pdf f_i(x) and CDF F_i(x), i=1, 2, ..., k. all independent. Define Y_i=F_i(X_i), i=1, ..., k. Derive the distribution of U=-2ln[Y_1.Y_2...Y_k].
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a....
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?
A continuous random variable X has cdf F given by: F(x)x3, x e [0,1] (1, x〉1...
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]
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...
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
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. Let X be a continuous r.v. with pdf fx() for which EX exists. We want to find the real number a that minimizes EX-al over all real a. Determine an equation involving the CDF Fx(-) that can be solved to obtain a*.
2. Let X be a continuous r.v. with pdf fx() for which EX exists. We want to find the real number a that minimizes EX-al over all real a. Determine an equation involving the CDF Fx(-) that...
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?
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]
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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