# 2S1: Assignment 4 ```2S1: Assignment 4
27th February 2009
Due date: Tuesday 31st March
1. A random variable X has a Poisson distribution with unknown parameter
λ. Given a random sample x1 , . . . , xn , show that the maximum likelihood
estimate for λ is the sample mean x̄.
2. Let X be a continuous random variable with an exponential distribution.
The probability density function for X is
λ e−λx if x &gt; 0
f (x) =
0
otherwise
where λ is an unknown parameter. Given a random sample x1 , . . . , xn , show
that the method of moments estimate for λ is x̄1 .
3. A population follows a normal distribution with unknown variance σ 2 . If
a random sample of size 20 has sample variance s2 = 1.1, find a 95%
confidence interval for the variance of the total population.
4. A scientist samples a chemical and finds that in 6 observations the levels of
phosporous are
3, 5, 7, 4, 5, 6
A second scientist takes a sample of 8 observations and obtains the results
8, 7, 9, 11, 7, 8, 10, 12
Carry out a test with a significance level of 5% to determine if there is a
significant difference between the population means. (The variance of both
populations are assumed to be the same).
5. Use a χ2 goodness of fit test to determine if the following data follows a
binomial distribution with p = 0.5:
x
frequency
0
5
1
6
2
9
1
3
7
4
2
```