Exercises and Exams

Question 1

Example 4.1 (Linear transformation in one dimension).
Y = aX + b, what is f_Y(y)?

Question 2

Example 4.6
Consider X,Y~N(0,1) and M = (X+Y)/2 and N+(X-Y)/2 such that (M,N) = g(X,Y). What is the joint distribution and the marginal distributions of M and N?

Question 3

Example (MLE for discrete uniform).
Bag contains unknown numbered balls.
Select k at random with replacement recording the label.
want to estimate n.

Question 4

Example (MLE for Simple linear regression).
Y_i = \gamma + \beta x_i + \sigma \varepsilon_i for iid \varepsilon ~N(0,1), with \sigma > 0.
What are MLEs for \gamma, \beta, \sigma.

Question 5

Example (MLE for Multinormal).
sample n individuals fropm a population of k types and observe n_i indidivudals of type k_i. wush to estimate the proportions of each type.

Question 6

Example 7.2 (Normal data with known variance and unknown mean).
suppose x_1, …, x_n are modelled by iid N(\mu, \sigma^{2}) with unknown \mu and known \sigma^2. Bias and MSE for estimator for \mu?
is it consistent?

Question 7

Example 7.3 (Uniformly dist. data with unknown range).
suppose x_1, \dots, x_n are ii.d Uniform (0, \theta) where \theta unknown.
MSE and Bias for estiamtor for theta?

Question 8

Example 8.1 (Estimation of normal mean when variance is known).
construct confidence interval for mean estiamtor if x_1, …, x_n are iid N(\mu, \sigma^{2}) with \sigma known.

Question 9

Example 8.2 (Estimation of RH range of uniform dist.)
Suppose x_1, …, x_n are ii.d Uniform (0, \theta). want 100(1-a)% confidenence interval for estiatmor for \theta.

Question 10

Example 8.3 (Normal estimation of variance)
x_1, …, x_n iid N(\mu, sigma^2) with \mu and \sigma to be estimated.
want confidence interval for \sigma^2.

Question 11

Example 8.5 (Normal estimation of mean when variance is unknown).
x_1, …, x_n iid N(\mu, \sigma).
interval for \mu.

Question 12

Example 8.6 (confidence interval for simple linear regression).

Question 13

Example (confidence interval for the binomial dist.)

Question 14

Example (multinomial confidence interval).

Question 15

Example 9.1 (one-sample t-test).

Question 16

Example (Hypothesis testing for simple linear regression)

Question 17

Example 9.3 (statistical test for categorial)

Question 18

1.A.2.b)
Find PMF of Y = floor(X/A) for X~Exp(\lambda)

Question 19

1.C.5.a)
X~N(0,1), Y=X^2, find distribution function of Y using \Phi.

b) hence find the pdf of Y.

Question 20

2.A.1
X~Multinomial(n, p), find M_X(t).

Question 21

2.C.1
X,Y~N(0,1) indpendent. Z = X^2 + Y^2. By integrating the joint density over the disc of radius r in polar coordinates, show that Z~Exp(?).