Question

In: Statistics and Probability

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

Expert Solution

Given: and that they are independent. Hence is also normally distributed.

Let

So, .

We want:

I did all the above manipulation because . Now we can write the probabilities in terms of the standard normal distribution .

Here is the CDF of the standard normal distribution Z.

orchestra answered 10 months ago

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)?

1. Let X and Y be independent U[0, 1] random variables, so that the point (X,...

1. Let X and Y be independent U[0, 1] random variables, so that the point (X, Y) is uniformly distributed in the unit square. Let T = X + Y. (a) Find P( 2Y < X ). (b). Find the CDF F(t) of T (for all real numbers t). HINT: For any number t, F(t) = P ( X <= t) is just the area of a part of the unit square. (c). Find the density f(t). REMARK: For a...

Let Xi's be independent and identically distributed Poisson random variables for 1 ≤ i ≤ n....

Let Xi's be independent and identically distributed Poisson random variables for 1 ≤ i ≤ n. Derive the distribution for the summation of Xi from 1 to n. (Without using MGF)

Let X; be n IID U(0, 1) random variables. What are the mean and variance of...

Let X; be n IID U(0, 1) random variables. What are the mean and variance of the minimum-order and maximum-order statistics? PLEASE SHOW ALL WORK AND FORMULAS USED

Let U, V be iid Unif(0, 1) random variables, and set M = max(U,V) and N...

Let U, V be iid Unif(0, 1) random variables, and set M = max(U,V) and N = min (U,V) (a) Find the conditional density of N given M = a for any value of a ∈ (0, 1). (b) Find Cov(M, N).

1. Let ?1, . . . ?? be ? independent random variables with normal distribution of...

1. Let ?1, . . . ?? be ? independent random variables with normal distribution of expectation 0 and variance ? 2 . Let ?̂︁2 1 be the sample variance ??, ?̂︁2 2 be 1 ? ∑︀ ? ?2 ? . (1) Show that the expectation of ?̂︁2 1 and ?̂︁2 2 are both ? 2 . In other words, both are unbiased point estimates of ? 2 . (2) Write down the p.d.f. of ?̂︁2 1 and ?̂︁2 2....

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f....

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f. ?(?) = ???−1, 0 ≤ ? ≤ 1 . c) Show that the maximum likelihood estimator for ? is biased, and find a function of the mle that is unbiased. (Hint: Show that the random variable −ln (??) is exponential, the sum of exponentials is Gamma, and the mean of 1/X for a gamma with parameters ? and ? is 1⁄(?(? − 1)).) d)...

Let ?1 and ?2 be independent normal random variables with ?1~ ?(µ, ? 2 ) and...

Let ?1 and ?2 be independent normal random variables with ?1~ ?(µ, ? 2 ) and ?2~ ?(µ, ? 2 ). Let ?1 = ?1 and ?2 = 2?2 − ?1. a) Find ??1,?2 (?1,?2). b) What are the means, variances, and correlation coefficient for ?1 and ?2? c) Find ?2(?2).

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.

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let...

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2, ?3}. Find the variance of ?2, and the covariance between the median ?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).