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. You may want to use the definition of ? 2 distribution. (3) Calculate ? ??(?̂︁2 2) ? ??(?̂︁2 1) .

Solutions

Expert Solution

NOTE:: I HOPE YOUR HAPPY WITH MY ANSWER......***PLEASE SUPPORT ME WITH YOUR RATING......THANK YOU

orchestra answered 1 year ago

Next > < Previous

Related Solutions

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).

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).

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

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 1. Write down the joint cumulative distribution function and joint probability density function of ?1 + ?2 and ?1?2. 2. Write down the covariance between ?1 + ?2 and ?1?2. 3. Let ? be the largest magnitude (absolute value) of a root of the equation ?^2 − ?1? + ?2 = 0. Let ? be the random event that says that the equation ?^2 −?1?...

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

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 1. Write down the joint cumulative distribution function and joint probability density function of ?1 + ?2 and ?1?2. 2. Write down the covariance between ?1 + ?2 and ?1?2. 3. Let ? be the largest magnitude (absolute value) of a root of the equation ? 2 − ?1? + ?2 = 0. Let ? be the random event which says that the equation ?...

Let {Zt} be independent normal random variables with mean 0 and variance σ2. Let a, b,...

Let {Zt} be independent normal random variables with mean 0 and variance σ2. Let a, b, c be constants. Which of the following processes are stationary? Evaluate mean and autocovariance function. (a) Xt = Ztcos(at) + Zt−1sin(bt) (b) Xt =a+bZt + cZt−2 (c) Xt = ZtZt−1

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

2. 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).

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2,...

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2, Y2 = Z1 − Z2. (a) Find the joint pdf fY1,Y2(y1,y2). Don’t use the change of variables theorem – all of that work has already been done for you. Instead, evaluate the matrices Σ and Σ−1, then multiply the necessary matrices and vectors to obtain a formula for fY1,Y2(y1,y2) containing no matrices and no vectors. (b) Find the marginal pdf fY2 (y2). Don’t use...

Xn are independent random variables and each one of them has a normal distribution with mean...

Xn are independent random variables and each one of them has a normal distribution with mean 0 and variance 1 what is the distribution of X-bar if n=9 what is the probability of X-bar =<3 if n=9

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)...

Subjects