Show that the covariance between $\hat { \alpha }$ and $\hat { \beta }$ is zero.

Let $\mathbf { X } ^ { \prime } = \left[ X _ { 1 } , X _ { 2 } , X _ { 3 } \right]$ denote a random sample of size 3 from a distribution that is N(4, 8) and let

$$\mathbf { A } = \left( \begin{array} { c c c } { \frac { 1 } { 2 } } & { 0 } & { \frac { 1 } { 2 } } \\ { 0 } & { 1 } & { 0 } \\ { \frac { 1 } { 2 } } & { 0 } & { \frac { 1 } { 2 } } \end{array} \right)$$

Let $Q = \mathbf { X } ^ { \prime } \mathbf { A } \mathbf { X } / \sigma ^ { 2 }$ (a) Use Theorem to find the E(Q). (b) Justify the assertion that Q is $\chi ^ { 2 } ( 2,6 )$

A person bets 1 dollar to b dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. Find b so that the bet is fair.

Players A and B play a sequence of independent games. Player A throws a die first and wins on a “six.” If he fails, B throws and wins on a “five” or “six.” If he fails, A throws and wins on a “four,” “five,” or “six.” And so on. Find the probability of each player winning the sequence.

Let $X_1,X_2, . . .,X_n$ be a random sample from a distribution with pdf $f ( x ; \theta ) = \theta \exp \left\{ - | x | ^ { \theta } \right\} / 2 \Gamma ( 1 / \theta ) , - \infty < x < \infty,$ where $\theta > 0.$ Suppose $\Omega = \{ \theta : \theta = 1,2 \}$ Consider the hypotheses $H _ { 0 } : \theta = 2$ (a normal distribution) versus $H _ { 1 } : \theta = 1$ (a double exponential distribution). Show that the likelihood ratio test can be based on the statistic $W = \sum _ { i = 1 } ^ { n } \left( X _ { i } ^ { 2 } - \left| X _ { i } \right| \right)$

Let $X_1,X_2, . . .,X_n$ and $Y_1, Y_2, . . . , Y_m$ be independent random samples from $N \left( \theta _ { 1 } , \theta _ { 3 } \right)$ and $N \left( \theta _ { 2 } , \theta _ { 4 } \right)$ distributions, respectively. (a) If $\Omega \subset R ^ { 3 }$ is defined by $\Omega = \left\{ \left( \theta _ { 1 } , \theta _ { 2 } , \theta _ { 3 } \right) : - \infty < \theta _ { i } < \infty , i = 1,2 ; 0 < \theta _ { 3 } = \theta _ { 4 } < \infty \right\}$ find the mles of $\theta _ { 1 } , \theta _ { 2 } ,$ and $\theta _ { 3 }$ (b) If $\Omega \subset R ^ { 2 }$ is defined by $\Omega \subset R ^ { 2 }$ is defined by $\Omega = \left\{ \left( \theta _ { 1 } , \theta _ { 3 } \right) : - \infty < \theta _ { 1 } = \theta _ { 2 } < \infty ; 0 < \theta _ { 3 } = \theta _ { 4 } < \infty \right\}$ find the mles of $\theta _ { 1 }$ and $\theta _ { 3 }$

Let $X_1,X_2, . . .,X_n$ be a random sample from a Poisson distribution with mean $\theta > 0$. Consider testing $H _ { 0 } : \theta = \theta _ { 0 }$ against $H _ { 1 } : \theta \neq \theta _ { 0 }$ (a) Obtain the Wald type test of expression. (b) Write an R function to compute this test statistic. (c) For $\theta _ { 0 } = 23$, compute the test statistic and determine the p-value for the following data. 27 13 21 24 22 14 17 26 14 22 21 24 19 25 15 25 23 16 20 19

Show that $R = \frac { \sum _ { 1 } ^ { n } \left( X _ { i } - \overline { X } \right) \left( Y _ { i } - \overline { Y } \right) } { \sqrt { \sum _ { 1 } ^ { n } \left( X _ { i } - \overline { X } \right) ^ { 2 } \sum _ { 1 } ^ { n } \left( Y _ { i } - \overline { Y } \right) ^ { 2 } } } = \frac { \sum _ { 1 } ^ { n } X _ { i } Y _ { i } - n \overline { X Y } } { \sqrt { \left( \sum _ { 1 } ^ { n } X _ { i } ^ { 2 } - n \overline { X } ^ { 2 } \right) \left( \sum _ { 1 } ^ { n } Y _ { i } ^ { 2 } - n Y ^ { 2 } \right) } }$

Show that the nth order statistic of a random sample of size n from the uniform distribution having pdf $f ( x ; \theta ) = 1 / \theta , 0 < x < \theta , 0 < \theta < \infty$ zero elsewhere, is a sufficient statistic for $\theta.$ Generalize this result by considering the pdf $f ( x ; \theta ) = Q ( \theta ) M ( x ) , 0 < x < \theta , 0 < \theta < \infty,$ zero elsewhere. Here, of course, $\int _ { 0 } ^ { \theta } M ( x ) d x = \frac { 1 } { Q ( \theta ) }$

Let X be a random variable with the pdf of a regular case of the exponential class, given by $f ( x ; \theta ) = \exp [ \theta K ( x ) + H ( x ) + q ( \theta ) ] , a < x < b , \gamma < \theta < \delta.$ Show that $E [ K ( X ) ] = - q ^ { \prime } ( \theta ) / p ^ { \prime } ( \theta ),$ provided these derivatives exist, by differentiating both members of the equality $\int _ { a } ^ { b } \exp [ p ( \theta ) K ( x ) + H ( x ) + q ( \theta ) ] d x = 1$ with respect to $\theta.$ By a second differentiation, find the variance of K(X).

Let $X_1,X_2, . . . , X_n, n > 2,$ be a random sample from the binomial distribution $b ( 1 , \theta )$ (a) Show that $Y _ { 1 } = X _ { 1 } + X _ { 2 } + \cdots + X _ { n }$ is a complete sufficient statistic for $\theta.$ (b) Find the function $\varphi \left( Y _ { 1 } \right)$ that is the MVUE of $\theta.$ (c) Let $Y _ { 2 } = \left( X _ { 1 } + X _ { 2 } \right) / 2$ and compute $E(Y_2).$ (d) Determine $E(Y_2|Y_1 = y_1).$

Let $X_1,X_2, . . . , X_n$ denote a random sample from a Poisson distribution with parameter $\theta > 0$ We know that $E \left[ ( - 1 ) ^ { X _ { 1 } } \right] = e ^ { - 2 \theta }$ (a) Show that $E \left[ ( - 1 ) ^ { X _ { 1 } } | Y _ { 1 } = y _ { 1 } \right] = ( 1 - 2 / n ) ^ { y _ { 1 } }$ where $Y _ { 1 } = X _ { 1 } + X _ { 2 } + \cdots + X _ { n }$ Hint: First show that the conditional pdf of $X _ { 1 } , X _ { 2 } , \ldots , X _ { n - 1 }$ given $Y_1 = y_1,$ is multinomial, and hence that of $X_1,$ given $Y_1 = y_1,$ is $b \left( y _ { 1 } , 1 / n \right)$ (b) Show that the mle of $e ^ { - 2 \theta }$ is $e ^ { - 2 \overline { X } }$ (c) Since $y _ { 1 } = n \overline { x }$ show that $( 1 - 2 / n ) ^ { y _ { 1 } }$ is approximately equal to $e ^ { - 2 \overline { x } }$ when n is large.

Let $X_1,X_2, . . . , X_n$ be a random sample from a Poisson distribution with mean $\theta > 0$ (a) Statistician A observes the sample to be the values $x_1, x_2, . . . , x_n$ with sum $y = \sum x _ { i }$ Find the mle of $\theta$ (b) Statistician B loses the sample values $x_1, x_2, . . . , x_n$ but remembers the sum $y_1$ and the fact that the sample arose from a Poisson distribution. Thus B decides to create some fake observations, which he calls $z_1, z_2, . . . , z_n$ (as he knows they will probably not equal the original x-values) as follows. He notes that the conditional probability of independent Poisson random variables $Z_1, Z_2, . . . , Z_n$ being equal to $z_1, z_2, . . . , z_n,$ given $\sum z _ { i } = y _ { 1 }$ is $\frac { \frac { \theta ^ { z _ { 1 } } e ^ { - \theta } } { z _ { 1 } ! } \frac { \theta ^ { z _ { 2 } } e ^ { - \theta } } { z _ { 2 } ! } \cdots \frac { \theta ^ { z _ { n } } e ^ { - \theta } } { z _ { n } ! } } { \frac { ( n \theta ) ^ { y _ { 1 } } e ^ { - n \theta } } { y _ { 1 } ! } } = \frac { y _ { 1 } ! } { z _ { 1 } ! z _ { 2 } ! \cdots z _ { n } ! } \left( \frac { 1 } { n } \right) ^ { z _ { 1 } } \left( \frac { 1 } { n } \right) ^ { z _ { 2 } } \cdots \left( \frac { 1 } { n } \right) ^ { z _ { n } }$ since $Y _ { 1 } = \sum Z _ { i }$ has a Poisson distribution with mean $n \theta$ The latter distribution is multinomial with $y_1$ independent trials, each terminating in one of n mutually exclusive and exhaustive ways, each of which has the same probability 1/n. Accordingly, B runs such a multinomial experiment $y_1$ independent trials and obtains $z_1, z_2, . . . , z_n.$ Find the likelihood function using these z-values. Is it proportional to that of statistician A? Hint: Here the likelihood function is the product of this conditional pdf and the pdf of $Y _ { 1 } = \sum Z _ { i }$

Let $X_1,X_2, . . . , X_n$ denote a random sample from a Poisson distribution with parameter $\theta , 0 < \theta < \infty .$ Let $Y = \sum _ { 1 } ^ { n } X _ { i }$ and let $\mathcal { L } [ \theta , \delta ( y ) ] = [ \theta - \delta ( y ) ] ^ { 2 }$. If we restrict our considerations to decision functions of the form $\delta ( y ) = b + y / n,$ where b does not depend on y, show that $R ( \theta , \delta ) = b ^ { 2 } + \theta / n$ What decision function of this form yields a uniformly smaller risk than every other decision function of this form? With this solution, say $\delta$ and $0 < \theta < \infty,$ determine $\max _ { \theta } R ( \theta , \delta )$ if it exists.