概率与统计代写-EM2B
时间:2022-03-28
EM2B coursework 2: Probability & Statistics
Answer all questions. You are strongly encouraged to use a consistent notation so that your
answers are easy to read and mark. Write clearly in good handwriting. Unless you are asked
explicitly, you do not need to derive from scratch any theorems we have covered in the class.
Limit your answer to a maximum of 7 A4 pages. Use only A4 size paper.
Question 1
A shipment of 1000 sacks of oranges arrives at a distribution centre in Edinburgh, one of
10 such facilities in Scotland. The sender quotes a nominal weight of 20 kg and a standard
deviation of no more than 2000 g per sack. A random sample of 10 sacks is selected and
weighed, yielding the data below in kg:n
21.7629, 18.1272, 19.7044, 22.1150, 16.8006, 23.0314, 22.9463, 23.1920, 20.2317, 19.7803
o
Throughout this question you should assume a Gaussian model for the data.
(a) (1)Write down the data likelihood function defining all variables you use, and hence calcu-
late the maximum likelihood estimator (MLE) for the population mean and variance.
How does the sample variance compared to the MLE estimator for the variance in this
particular case?
(b) (2)Assume that the true standard deviation is (as claimed by the sender) is 2000 g and
perform a hypothesis test regarding the claim that the average sack weight is 20 kg
using the above data at a level of significance ↵ = 0.05 and compute the p value of the
test. Further compute the power of the test if the true mean is µ1 = 22.
(c) (2)The vendor claims that the standard deviation in the weight of the sacks is less than 2
kg. Construct an appropriate confidence interval at confidence level 1 ↵ = 0.95 for
the variance and test the vendor’s claim.
Question 2
Let X1, . . . , Xn be a collection of random variables with joint density pX(x1, . . . , xn) and let
g : Rn ! Rn be an invertible function from (x1, . . . , xn) to (y1, . . . , yn) like
Yi = gi(X1, . . . , Xn) for i = 1, . . . , n.
The transformed variables Y1, . . . , Yn have a joint density pY (y1, . . . , yn) given by
pY (y1, . . . , yn) = pX(g
1(y1, . . . , yn))|detrg1(y1, . . . , yn)|.
In the above, | detA| is the absolute value of the determinant of the matrix A, and rf
denotes the Jacobian of the function f .
(a) (3)Let n = 2 and assume that X1, X2 ⇠ Normal(0, 1) are two independent standard normal
random variables, and consider the following parameters as known: 1 < µ1, µ2 <
+1, 0 < 1,2 < +1 and ⇢ such that 1 < ⇢ < 1. Now consider the function
g : R2 ! R2
g(x1, x2) =

1x1 + µ1,2(⇢x1 +
p
1 ⇢2x2) + µ2

For (Y1, Y2) = g(X1, X2) derive the joint density pY (y1, y2) of the pair Y1, Y2 in its
simplest form and compute their covariance.
SCEE08010 Engineering Mathematics 2B
(b) (2)For the variables Y1, Y2 as in the previous part consider the conditional density function
of Y2 given Y1 = y1. For this distribution work out the conditional mean and variance.
SCEE08010 Engineering Mathematics 2B


essay、essay代写