t x 1 e t dt, and simplify the answer when possible (for example, when r is a positive even number). In particular, confirm that EX 4 = 3.

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1 Mathematical Statistics: Homewor problems General guideline. While woring outside the classroom, use any help you want, including people, computer algebra systems, Internet, and solution manuals, but mae sure you are ready for exams, where you are on your own. The final project. [The project is due as two pdf files by Friday, April 7, 08, at noon ( to There are two parts: ( Solve one of the previous 54a qualifying exams, write the solution carefully, and organize it as a single pdf file. Mae sure to include the statement of each problem. ( Mae your own qual for math 54a, and then solve it. Submit the result as another pdf file. Collaborations are welcome, but everybody should prepare and submit their own wor. Different people should solve different quals for the first part and thin of different problems for the second part. Homewor. Problem. [Probability integral transformation, Th...0 in the boo] Let X be a random variable with a continuous cdf F X = F X (x. Define the random variable Y by Y = F X (X. Show that Y is uniform on [0, ], that is, P(Y y = y, y (0,. Problem. Give an example of a random variable X with each of the following properties ( E X ε = + for every ε > 0; ( E X r ε < for some r > 0 and every 0 < ε < r, but E X r = + ; (3 E X r < for some r > 0, but E X r+ε = + for every ε > 0; (4 E X r < for every r > 0, but Ee λx = + for every λ R. In each case, you need to come up with a suitable pdf or pmf or cdf. Problem 3. Let X be a standard Gaussian random variable. Determine the values of the real number r for which E X r exits and compute the expectation for those r. Express your answer in terms of the Gamma function Γ(x = + 0 t x e t dt, and simplify the answer when possible (for example, when r is a positive even number. In particular, confirm that EX 4 = 3. Problem 4. Let X be a random variable with E X 4 <. Define µ = EX and µ = E(X µ, =, 3, 4, and then α 3 = µ 3 µ 3/ (sewness, α 4 = µ 4 µ (urtosis ( Show that if P(X µ > x = P(X µ < x for every x > 0, then µ 3 = 0, but not the other way around. ( Compute α 3 and α 4 when X is Binomial with parameter p, exponential with mean, uniform on [0, ], standard normal, and double exponential (f X (x = (/e x. Problem 5. (a Let f (x = x π e (ln x /, f (x = f (x ( + sin(π ln x, x > 0. Confirm that both f and f are probability density functions and, for every = 0,,,..., 0 x f (xdx = 0 x f (xdx = e /.

2 (b Let X be a standard normal random variable, and let be a positive integer. Show that the random variable X is uniquely determined by its moments if and only if =,, 4. [Reference: C. Berg, The cube of a normal distribution is indeterminate, Ann. Probab., 6(, 90 93, 988] Homewor. Problem. Consider the hypergeometric distribution with pmf p (M, N, K = ( M ( N M K ( N, = 0,..., K. K Confirm that lim p (M, N, K = M,N,M/N p ( K p ( p K, = 0,..., K; lim p (M, N, K = e λ λ, = 0,,,... M,N,K,MK/N λ! Problem. Given a random variable X with possible values 0,,,... and pmf p (X, there could be two ways to define a new random variable that models the situation when the values X = 0 cannot be observed: ( Zero-truncated random variable X T with pmf p (T = p (X p(x, =,, 3,... ( Size-biased random variable X S with pmf p (S = p (X p(x, =,, 3,... Compute p (T, EX T, and VarX T when p (X = e λ λ Compute p (S, EX S, and VarX S when p (X = e λ λ Compute p (T! ;! ;, EX T, and VarX T when p (X = ( N+ p N ( p ; Compute p (S, EX S, and VarX S when p (X = ( N+ p N ( p ; Compare and contrast zero truncation and size biasing. Use your judgement as to what and how much to write. Problem 3. Let X gamma(α, β. Determine all the values of r R for which EX r < and express EX r for those values of r in terms of r, α, β, and the Gamma function. Problem 4. For each of the following families of distributions (defined either by a pdf or a pmf or by a transformation of another family determine whether it is an exponential family and/or location-scale family. If it is an exponential family, then identify the natural

3 3 parameter space. ( N p = p ( p N, = 0,,..., N, N fixed (B(N, p ( N + p = p N ( p, = 0,,..., N fixed p = e λ λ, = 0,,..., λ > 0! ( p p =, = 0,,... ln(/p e (x µ, x > µ βαβ x β+ exp π σ ( e (x θ ( + e (x θ σ e x µ /σ (x µ σ (N (µ, σ a θ a xa, x (0, θ, a > 0 ( λ λ(x µ πx exp, x > 0 3 µ x B(α, β xα ( x β, x (0, Γ(α, β xα e x/β (gamma(α, β π σ( + (x µ /σ ( /γ gamma(, β /gamma(α, β ( α γ ln gamma(, (beta(α, β Problem 5. For each of the families in the previous problem, determine whether the family is stochastically monotone [that is, either stochastically increasing or stochastically decreasing] with respect to each of the parameters. Problem 6. Let Z be a standard normal random variable. Confirm that, for every x > 0, π ( x / + x e x P( Z x min, π e x /. x Homewor 3. Problem. (a Let X P(λ and Y P(µ be independent Poisson random variables. Compute the conditional distributions of X and Y given X + Y.

4 4 (b Let X and Y be iid geometric. Confirm that min(x, Y and X Y are independent; determine the distribution of X/(X + Y (under the convention 0/0 = 0; compute the joint distribution of X and X + Y. (c Let g = g(t, t > 0 be a non-negative function with g(tdt <. With r = x 0 + y, determine the value of c such that f(x, y = g(r/r is a pdf and confirm that if X, Y are random variables with joint pdf f, then X/Y has Cauchy distribution. (d Confirm that if X gamma(α, λ and Y gamma(β, λ are independent, then X+Y gamma(α + β, λ, X/(X + Y beta(α, β, and X + Y and X/(X + Y are independent. (e Confirm that if X, Y are iid standard normal, then X + Y is exponential, tan (Y/X is uniform, and X + Y and tan (Y/X are independent. Conversely, confirm that if U and U are iid uniform on (0,, then ln U cos(πu and ln U sin(πu are iid standard normal; this is nown as the Box-Muller transformation and is used to generate normal random variables. Problem. (a Consider the hierarchical model Y Λ P(Λ, Λ gamma(α, β Compute the unconditional distribution of Y (maing sure it is negative binomial if α is integer, as well as the expectation and variance of Y. (b Confirm that the hierarchical model Y N B(N, p, N Λ P(Λ, Λ gamma(α, β leads to the same unconditional distribution of Y as in part (a. Problem 3. Consider the hierarchical model X i p i B(n i, p i, p i beta(α, β, i =,..., N, with n i nown and non-random. Define Y = N i= X i. Assuming all the independence you need, compute the mean and the variance of Y and, if possible, also the unconditional distribution of Y. Problem 4. The multi-dimensional version of the beta distribution is called the Dirichlet distribution and is constructed as follows: define the set S n = { n x R n : x i 0, x i = } and, for fixed α i > 0, the function n C x α i i, x S n ; 0 otherwise. i= Once C is chosen so that S n f(xdx =, the random vector X = (X,..., X n with joint pdf f is said to have the Dirichlet distribution. ( Determine the value of C so that f is a pdf; ( Compute the marginal distribution of each X i ; (3 Compute the conditions distribution of X i given X j. (4 Compute the covariance Cov(X i, X j. Problem 5. Verify the following properties of the conditional expectation. When in doubt, you are welcome to consider only the particular case when the random variables X, Y have a joint density or pmf. ( E ( E(X Y = EX; ( E ( g(y X Y = g(y E ( X Y for every bounded measurable function g; (3 Cov ( E(X Y, Y = Cov(X, Y ; (4 Y and X E(X Y are uncorrelated; (5 E ( X Y = EX if X and Y are independent. i=

5 Problem 6. (a Let X and Y be independent random variables with pdfs f X and f Y. Express the pdfs of the following random variables in terms of f X and f Y : ax + by (a, b R, XY, X/Y. (b Let X, Y, Z be independent uniform on (0,. Compute the cdfs of XY, X/Y, and XY/Z. Problem 7. (a Let X be standard normal, a > 0, and define { X, if X > a Y = X, if X a Identify the distribution of Y, compute Cov(X, Y as a function of a, and show that Cov(X, Y = 0 for some a. For what values of a with the vector (X, Y be jointly normal? (b Let X and Y be iid standard normal and define { X, if XY > 0 Z = X, if XY 0 Confirm that Z is normal but (Z, Y are not jointly normal. (c Let X, Y, Z be iid standard normal. Confirm that V = X + Y Z + Z is also standard normal. Is any of the vectors (X, V, (Y, V, (Z, V jointly normal? (d Let X, Y be standard normal such that the joint distribution of X and Y is also normal and EXY = ρ. Compute E(X Y and then the correlation coefficient between X and Y. Problem 8. Below, i =, (, is inner product in Euclidean space; C means inverse of the matrix C; C T means the transpose of C. Vectors are thought of as matrices with one column. (a Confirm that the following three definitions of a Gaussian vector X = (X,..., X n are equivalent: ( Ee i(x,λ = e i(λ,µ (/(Cλ,λ for some vector µ and a symmetric non-negative definite matrix C; with this characterization, also confirm that µ = EX and C = C XX is the covariance matrix of X; ( (a, X is a Gaussian random variable for every a R n (3 X = GZ, where G is a symmetric non-negative definite matrix and Z is a vector with iid standard normal components. (b Let X be a Gaussian vector in R n, let Y be a Gaussian vector in R m and assume that the combined vector X, Y is Gaussian in R m+n and the covariance matrix C Y Y of Y is invertible. Confirm that E(X Y = EX +C XY C Y Y (Y EY, E (X E(X Y Note that C Y X = C T XY. Start by finding a matrix A such that the vector X EX A(Y EY ( T X E(X Y = CXX C XY C Y Y C Y X. and the vector Y EY are uncorrelated. [Hint: A = C XY C Y Y ]. Confirm that if m = n =, then the conditional expectation is the equation of the regression line of Xon Y. 5

6 6 Homewor 4. Problem. Confirm that, for the sample mean X n and the (unbiased sample variance Sn, X n+ = X n X ( n X n+ n, Sn+ = S n + ( X n X n+. n + n n + Problem. (a Compute the mean and the variance of χ n and confirm that χ n is stochastically increasing in n. (b Compute the mean and the variance of t n and confirm that, as n, t n N (0, in distribution. (c Compute the mean and the variance of F m,n, confirm that mf m,n is stochastically increasing in m for fixed n, and, with γ = n/m, the random variable γ/(f n,m + γ has beta(m/, n/ distribution. Problem 3. (a Let X, X,... be iid exponential with mean λ. Confirm that the random variables X (, X ( X (,..., X (n X (n are independent and each has exponential distribution. Compute the expected values of those random variables. (b Confirm that, in a Poisson process conditioned on N events happening over the fixed time interval [0, T ], the times of events are the order statistics of iid random variables that are uniform on [0, T ]. (c Let U,..., U n be iid uniform on (0,, let X,..., X n+ be iid exponential with mean one, and define Y = X, Y = Y +X, =,..., n+. Confirm that the joint distribution of the order statistics U (,..., U (n is the same as the joint distribution of the random variables Y /Y n+,..., Y n /Y n+. (d Let X,..., X n be iid with pdf aθ a x a (0 < x < θ. Confirm that the random variables X ( /X (, X ( /X (3,..., X (n /X (n, and X (n are independent and identify their (marginal distributions. (e Let X, X,... X n be iid with a continuous cdf. True or false: if E X ( < for some >, then E X (m < for all m <? [Hint: false; thin Cauchy] Problem 4. (a Let X,..., X n be iid beta(α, β. In what sense can you show that lim n X ( = 0 and lim n X (n =? How can you re-normalize X ( and X (n to get non-degenerate limits in distribution? [For example, for beta(, (that is, uniform we have n( X (n converging in distribution to the mean-one exponential. (b Let X,..., X n be iid exponential with mean one. In what sense can you show that lim n X ( = 0 and lim n X (n =? How can you re-normalize X ( and X (n to get non-degenerate limits in distribution? [For example, nx ( is exponential with mean one and X (n ln n converges to a version of the extreme value distribution.] Problem 5. Write carefully the examples showing that there is no connection between convergence almost sure and in L and that almost sure convergence does not follow from convergence in probability. Homewor 5. Problem. For each of the distributions in Problem 4 of Homewor, identify sufficient statistics and chec for minimality and completeness. If there are more than one parameter, do it for each parameter individually and then for all of them. [Try to do as many distributions as you can, and then some, especially if you are planning to do any ind of a qual: the distributions you ignore might be waiting for you there...] Problem. In each case, confirm that the given statistic T is sufficient for θ and chec whether it is complete and minimal. If not, see if you can find one or show that it does not

7 exist. Identify the examples that give you two our of three (minimal and sufficient but not complete or complete and sufficient but not minimal. ( X,..., X n are iid uniform on (θ, θ +, T = {X (, X (n }; ( X,..., X n are iid uniform on (θ, θ, T = {X (, X (n }; (3 X,..., X n iid N (θ, θ, T = { X n, Sn}; (4 X,..., X n iid N (θ, θ T = n = X. Problem 3. In each case confirm that the given statistic T is ancillary for θ. ( X, X are iid with pdf θx θ e xθ, T = ln X / ln X ; ( X, X are iid with pdf (/θf(x/θ, where f is a pdf [scale family], T = X /X ; (3 X,..., X n are iid with pdf f(x θ where f = f(x is a pdf [location family]; T = M n X n, M n is the sample median. Homewor 6. Note: The Statistics A quals you need for this homewor (and pretty much all other math quals, some going bac to previous century are posted here: Problem. Spring 07 qual. Problem. Fall 05 qual. Problem 3. Spring 04 qual. Problem 4. Problem from Fall 03 qual, Problem from, Fall 0 qual. Problem 5. [You new it was coming] For each distribution in Problem 4 of Homewor compute the method of moments and MLE of the parameters [in case of several parameters, do each one individually and then all at once], construct the UMVUE or explain why it does not exist, compute the Fisher information, and see if the lower bound in the Cramér-Rao inequality can be achieved. Do your best and try to do as much as you can. 7