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) .

Expert Solution

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

orchestra answered 1 year ago

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).

(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 ?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

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

Question

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

Solutions

Expert Solution

Related Solutions

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

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

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

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....

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

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

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

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

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