## Transcribed Text

Q1 One way to generate a continuous random variable X with a given distribution function
(edf) Fx(x) is to generate a uniform random variable U ~ U(0,1) and then define
x = Fx(U). Consider
In(x°+z+1)
In3
< x < 1,
Qli Fx (x; A) =
0,
otherwise
Here. 0 0 18 an unknown parameter.
(a) There is no simple formuls for the inverse edl Fx in this case, as it is usually
the case for many distributions We can use R to compute Fx'(x.0) numerically.
For 0 < q < 1 and 0 0. write a function in R that returns I that gives the
of the following objective function:
9(I.4.6) - ((xx(x)0)-4)2,
You can use function optin or nlm in R. The minimum should be zero and the
value x that minimites this functioni is x = Fx 1(g:8). Use this function to generale
a sample of size n = 100 from the edf Fx(x;8) with 8 = 2. Include the histogram
of the generated data in your report.
(b) The maximum likelihood estimate cannot be obtained in a simple form for this
distribution so we call use numerical methods Write a function in R that finds the
maximium likelihood estimate for the generated data n. 2100-
The function
should return 0 <8<1 that maximizes the log likelihood
6(I,
7100 @)
==1
where (x(x:0) is the pdf of X. Again, you can use funition optim or aln to lind
the Include the estimate in your report.
Q2 Consider the linear regression model:
Q2c
when I- In are some constants and Em are independent random vari-
ables and & ~ N(1,02).
(a) Generate a sample of size 12 = 20 from this model, Y1, Y2, with or = 1, 8 = 3
and of = 0.5 and I, = i/10, i = 1. 20. Plot the values Y against Ii
(b) Write a function in R that finds the maximum likelihood estimate of a.B and o'
for the gemerated data 3/1-
1/20 The function should return n. B and of > 0
that maximise the log likelihood
where is the density of Y,
(c) What is the predicted value of response variable Y if x = 3?

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# Q1

## (a)

Fx <- function(x, theta) {

log(x^theta+x+1) / log(3)

}

fx <- function(x, theta) {

(theta*x^(theta-1)+1) / (x^theta+x+1) / log(3)

}

g <- function(x, q, theta) {

Fx <- log(x^theta+x+1) / log(3)

(Fx - q)^2

}

invFx <- function(q, theta) {

output <- optim(par=0.5,

fn=g, q=q, theta=theta, method="Brent",

lower=0, upper=1000)

return(output$par)

}

n <- 100

theta <- 2.0

# generate random outcomes

x <- vector(length=n)...