Question 1. Let Xi, i = 1, ... ,50 be independent random variables with Xi having exponential distribution with mean (0.9)i.
(a) Calculate the mean and variance of X¯.
(b) Use simulation to estimate p(X¯ > 0.5) with 2 decimals precision.
(c) Does X¯ have approximately normal distribution? Justify your answer by simulating X¯ and drawing a histogram.
2. Let (Xi, Yi), i = 1,...., n be a random sample from a bivariate standard normal distribution with correlation coefficient p. Let S denote the sample covariance matrix and let Al and A2 denote the eigenvalues of S.
For n = 30, 40, 50 and ρ = 0.2, 0.5, 0.8 find the following:
(a) The marginal distribution of the λ1 and λ2.
(b) The means, variances and covariances of the eigenvalues of S
(c) Find the distribution of the max(λ1, λ2).
3. Let Y1.....Yn be a set on independent Poison random variables with the mean
λi = eβo + β1x1i + β2x2i
(a) Based on the following observations, write a function to find the maximum likelihood estimations for β0, β1 and β2.
| Y |
X1 |
X2 |
Y |
X1 |
X2 |
| 0 |
3 |
1 |
0 |
3 |
5 |
| 2 |
5 |
5 |
0 |
3 |
3 |
| 2 |
2 |
4 |
0 |
3 |
2 |
| 1 |
2 |
4 |
2 |
5 |
4 |
| 2 |
2 |
3 |
0 |
3 |
5 |
| 0 |
3 |
3 |
2 |
5 |
3 |
| 1 |
4 |
2 |
1 |
4 |
4 |
| 3 |
5 |
3 |
1 |
2 |
2 |
| 3 |
1 |
4 |
1 |
4 |
3 |
| 0 |
3 |
4 |
3 |
5 |
2 |
| 0 |
3 |
I |
0 |
3 |
1 |
| 2 |
5 |
3 |
2 |
5 |
4 |
| 1 |
2 |
3 |
0 |
3 |
1 |
| 0 |
3 |
1 |
2 |
2 |
3 |
| 2 |
2 |
1 |
0 |
3 |
2 |
(b) Assume that β0 = 0.1, β1 = 0.2 and β2 = 0.3 and matrix X is generated by R code: set . seed (890)
x < - cbind(sample(1:5,30,replace,-.T,c(.05,.2,.35,.25,15))
sample(1:5,30,replace=T.c(.15,.2,.3,.2,15)))
Write a function to generate independent Poison random variables Yi, i = 1,......30, 10000 times and returns a matrix of the maximum likelihood estimators of βi's.
(c) Using the function obtained in (a) find the means and variances of the βi's estimations.
(d) draw the empirical distributions for βi's estimations.
(e) Create a vector of predictions of Y when x1 = 3 and x2 = 5 using the rows of the matrix obtained in (a), then find the mean and the variances of the predictions.
(f) Draw a histogram of predictions.
4. Write a function that takes as input a vector x, then returns as output the vector (m,lb,ub), where m is the mean and (lb, ub) is a 95% confidence interval for it. That is m = x¯ - 1.96s/√n, lb = x¯ + 1.96s/√n, where s is standard deviation of the sample.
Then, write a program that applies your function to the following sample
11, 52. 87, 45, 39. 95, 42, 38, 10, 03, 48, 56.
To four decimal places you should be getting
(43.83333, 27.9526, 59.7140)
5. Here is some code for simulating a random variable. Write down the cdf of the random variable being simulated. Check your answer by plotting an empirical cdf from the simulated data, then plotting the actual cdf on the top.
sim < - function () {
U < - runif(1)
if ( U < 0.5 )
{ return(0)
} else {
return(- log((1 - U)/20))
}
}