Homework Answers
![\text{Let's find the CDF of X first. Let }D\text{ be the domain of }X\\ \text{Because the density exists, WLOG, we can assume D is an open set.}\\ \text{Then, for }x\in D\\ P(X\leq x)=P(F^{-1}(U)\leq x)=P(U\leq F(x))=F(x)\\ \text{[Because }U\sim U(0,1)]\\ \implies X\text{ has density given by: }\\ \frac{\partial }{\partial x}P(X\leq x), x\in D\\ =\frac{\partial }{\partial x}F(x), x\in D\\ =f(x),x \in D\\ \text{Thus, }X\text{ has density }f\text{ (Proved.)}.](http://img.homeworklib.com/questions/9c1240d0-927e-11ec-9134-7fa3622e467f.png?x-oss-process=image/resize,w_560)
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Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that...
Assume the continuous random variable X follows the uniform [0,1] distribution, and define another random variable...
Assume the continuous random variable X follows the uniform
[0,1] distribution, and define another random variable
We were unable to transcribe this imagea) Determine the CDF of Y. Hint: start by writing P(Y ), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
Problem 3: Assume the continuous random variable X follows the uniform[0,1] distribution, and define another random...
Problem 3: Assume the continuous random variable X follows the uniform[0,1] distribution, and define another random variable Y- In () 1-X a) Determine the CDF of Y. Hint: start by writing P(Y y), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
7. Suppose the random variable U has uniform distribution on [0,1]. Then a second random variable...
7. Suppose the random variable U has uniform distribution on [0,1]. Then a second random variable T is chosen to have uniform distribution on [O, U] Calculate P(T > 1/2)
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U...
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U F (X). Show that, as long as F(x) is strictly monotonic increasing, U is uniformly distributed on (0,1). Discuss why this result is important, given that it is known how to simulate Uniformly distributed random variables easily.
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ unifo...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let Calculate analytically the variance of X. (HINT: E g(x)f(x)dx, and the p.d.f. 10SzSI 0 o.t.w. f(x) of a uniform distribution is f(x) =
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]
Let X ~ U[0,1] be a standard uniform random variable. Find the probability density functions (pdf's)...
Let X ~ U[0,1] be a standard uniform random variable. Find the probability density functions (pdf's) of the following random variables: iii) Y = 1/x0.5
(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).
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...
Assume the continuous random variable X follows the uniform
[0,1] distribution, and define another random variable
We were unable to transcribe this imagea) Determine the CDF of Y. Hint: start by writing P(Y ), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
Problem 3: Assume the continuous random variable X follows the uniform[0,1] distribution, and define another random variable Y- In () 1-X a) Determine the CDF of Y. Hint: start by writing P(Y y), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
7. Suppose the random variable U has uniform distribution on [0,1]. Then a second random variable T is chosen to have uniform distribution on [O, U] Calculate P(T > 1/2)
2. Let X be a continuous r.v. with pdf f () and cdf F(x). Let U F (X). Show that, as long as F(x) is strictly monotonic increasing, U is uniformly distributed on (0,1). Discuss why this result is important, given that it is known how to simulate Uniformly distributed random variables easily.
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let Calculate analytically the variance of X. (HINT: E g(x)f(x)dx, and the p.d.f. 10SzSI 0 o.t.w. f(x) of a uniform distribution is f(x) =
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]
(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).
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...
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