STAT 417 Statistical Theory Midterm, Spring 2022
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Statistical Theory (STAT 417)
Midterm, Spring 2022
1. (15 points, 1 point each). Fill in the blanks.
(a) (3 points). Let X be a random variable with E(X) = 2.7 and V(X) = 3.4. Let Y = -2X . Then, E(Y ) = , V(Y ) = , Cov(X, Y ) = .
(b) (3 points). Let the PDF of X be f (x) = c if 1 < x < 5. Then, c = , E(X) = , and V(X) = .
(c) (3 points). Let (X, Y ) be bivariate random vector. Assume that the conditional PDF of X given Y is fx|Y(xly) = 2x/y2 for 0 < x < y and the marginal PDF of Y is fY (y) = 3y2 for 0 < y < 1. Then, the joint PDF of (X, Y ) is f (x, y) = for 0 < x < y < 1, the marginal PDF of X is fx(x) = for x e (0, 1), and E(X) = .
(d) (3 points). Let X1 , . . . , X20 be iid N (0, 3). Let Y = ∑i(2)
1 Xi(2). Then, Y ~ , E(Y ) = , and V(Y ) = .
(e) (3 points). Flip a balanced coin 2000 times. Let X be the total number of heads observed.
Then, E(X) = , V(X) = , and P (X < 1050) s .
2. (8 points). Let x = (X1, X2 , X3 , X4 )⊤ be a four dimensional normal random vector with
μ = E(x) = 
1(0) 
, Σ = Cov(x) = 
.
1 
0 0 0 5 
Let
y = 
) .
(a) (2 points). Provide the correlation matrix of x.
1 1/,3 1/,8 0 
corr(x) = 
.
(b) (2 points). Find the distribution of y .
(c) (2 points). Construct a χ3(2)-distribution using X2 , X3, and X4. Show your work.
(d) (2 points). Find a and b such that X1 + aX2 + bX3 is independent of X3 and X4 .
3. (4 points). The following problems are unrelated.
(a) (2 points). Let X ~ Poisson(λ). By Chebyshev’s inequality, show that X/λ -(P) 1 as λ - o.
(b) (2 points). Let X1 , . . . , Xn be iid with a common PMF given by
|
1 |
2 |
3 |
4 |
5 |
|
0.1 |
0.3 |
0.2 |
0.3 |
0.1 |
Compute the probability of P (
> 3.05) when n = 102 , 103 , 104, respectively.
4. (8 points). Solve maximum likelihood estimator (MLE) in the following problems. Show your work.
(a) (2 points). Let X1 , . . . , Xn be iid with a common PDF given by f9 (x) = θ 2 xe−9z for
θ > 0 and x > 0. Find the MLE of θ .
(b) (2 points). Let X1 , . . . , Xn be iid with a common PDF given by f9 (x) = θx−(9+1) for
θ > 0 and x > 1. Find the MLE of θ .
(c) (2 points). Let X ~ Bin(n, θ), where only X and n are observed. Find the MLEs of E(X) and V(X).
(d) (2 points). Let X1 , . . . , Xn be iid a common PDF f9 (x) = 2x/θ2 if 0 < x < θ and f9 (x) = 0 otherwise. Find the MLE of θ .
2022-04-26