Let U be distributed as Uni(0,1). Find the density of U^n for n = 1, 2,...

Let U be distributed as Uni(0,1). Find the density of U^n for n = 1, 2, 3.... and also for n = -1, -2, ....

Expert Solution

orchestra answered 3 years ago

Suppose U is uniform on (0,1). Let Y = U(1 − U). (a) Find P(Y >...

Suppose U is uniform on (0,1). Let Y = U(1 − U). (a) Find P(Y > y) for 0 < y < 1/4. (b) differentiate to get the density function of Y . (c) Find an increasing function g(u) so that g(U ) has the same distribution as U (1 − U ).

Let Z~N (0,1). Find the 76th percentile.

Let X1 and X2 be uniformly distributed in the region [−1,1]×[0,1]∪(1,2]×(1,3]. 1. Find Joint and Marginal...

Let X1 and X2 be uniformly distributed in the region [−1,1]×[0,1]∪(1,2]×(1,3]. 1. Find Joint and Marginal pdf of X1 and X2. 2.Find V (X1 + 3X2) ，( I‘ve asked this question already, but the person didn't give me the correct answer, so please, dont waste my question, if you dont know how to answer this question.)

Chapter P, Section 5, Exercise 134 Find the specified areas for a N(0,1) density. (a) The...

Chapter P, Section 5, Exercise 134 Find the specified areas for a N(0,1) density. (a) The area below z=0.9 Round your answer to four decimal places. (b) The area above z=1.3 Round your answer to four decimal places. (c) The area between z=1.78 and z=1.21 Round your answer to four decimal places.

6. Let X1, X2, ..., X101 be 101 independent U[0,1] random variables (meaning uniformly distributed on...

6. Let X1, X2, ..., X101 be 101 independent U[0,1] random variables (meaning uniformly distributed on the unit interval). Let M be the middle value among the 101 numbers, with 50 values less than M and 50 values greater than M. (a). Find the approximate value of P( M < 0.45 ). (b). Find the approximate value of P( | M- 0.5 | < 0.001 ), the probability that M is within 0.001 of 1/2.

(A derivation of the bivariate normal distribution) let $Z_{1}$ and $Z_{2}$ be independent n(0,1) random variables,...

(A derivation of the bivariate normal distribution) let $Z_{1}$ and $Z_{2}$ be independent n(0,1) random variables, and define new random variables X and Y by\\ \begin{align*} X=a_{x}Z_{1}+b_{x}Z_{2}+c_{x} \quad Y=a_{Y}Z_{1}+b_{Y}Z_{2}+c_{Y}\\ \end{align*} where $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ are constants.\\ if we define the constants $a_{x},b_{x},c_{x},a_{Y},b_{Y},c_{Y}$ by\\ \begin{align*} a_{x}=\sqrt{\frac{1+\rho}{2}}\sigma_{X}, b_{x}=\sqrt{\frac{1-\rho}{2}}\sigma_{X}, c_{X}=\mu_{X},\\ a_{Y}=\sqrt{\frac{1+\rho}{2}}\sigma_{Y}, b_{Y}=-\sqrt{\frac{1-\rho}{2}}\sigma_{Y},c_{Y}=\mu_{Y}\\ \end{align*} where \mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho are constants\\ \\ \begin{itemize} \item Question 1):\\ Show that $(X,Y)$ has the bivariate normal pdf with parameter\\s $\mu_{X}, \mu_{Y},\sigma^2_{x},\sigma^2_{Y},\rho$\\ \item Question 2):\\ if we start with bivariate normal parameters...

Let X1, X2 ,X3 ∼U[0,1] be independent, and letX(1),X(2),X(3) be the order statistics, i.e., X(1) <...

Let X1, X2 ,X3 ∼U[0,1] be independent, and letX(1),X(2),X(3) be the order statistics, i.e., X(1) < X(2) < X(3) (a) Find f(1)(x), f(2)(y) and f(3)(z). (b) Verify that the joint density of X(1) and X(2) is f(1,2)(x,y)=6(1−y), 0 (c) Find the conditional density of X(1) given X(2). (d) Find E(X(1)+X(3 )/ X(2))

Let (Un, U, n>1) be asequence of random variables such that Un and U are independent,...

Let (Un, U, n>1) be asequence of random variables such that Un and U are independent, Un is N(0, 1+1/n), and U is N(0,1), for each n≥1. Calculate p(n)=P(|Un-U|<e), for all e>0. Please give details as much as possible

Let Y 1 ,...,Y n be a sample from the density f(y) = λ 2 ye...

Let Y 1 ,...,Y n be a sample from the density f(y) = λ 2 ye −λy , y > 0 where λ > 0 is an unknown parameter. (a) Find an estimator 'λ 1 of λ by Method of Moments (b) Find an estimator 'λ 2 of λ by Method of Maximum Likelihood. (c) Find an estimator 'λ 3 of λ that is a Sufficient estimator. Can you construct a Minimal Variance Unbiased Estimator? Justify.

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let...

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let X = max(U, V). What is Cov(X, U)?

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