Download Tutorial 1 - City University of Hong Kong

MA3518: Applied Statistics
Page 1
Department of Mathematics
Faculty of Science and Engineering
City University of Hong Kong
MA 3518: Applied Statistics
Tutorial 1
Question 1:
Let M and N be two independent Poisson random variables of means  and
 , respectively, and R = M + N
(a) Prove that P(R = 0) = e-(  +  ) and P(R = 1) = (  +  ) e-(  +  )
(b) Show that R follows a Poisson distribution with mean (  +  )
(b) What is the conditional distribution for M given that R is equal to r?
Justify your answer.
Question 2:
Suppose the random variable X follows a binomial distribution with
parameters n and p (i.e. X ~ Bin(n, p))
(a) Find the mean and the standard deviation of X
(b) Calculate the skewness of X when n = 10, p = 0.3
(c) Describe the behavior of the skewness of X when p increases
MA3518: Applied Statistics
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Question 3:
Let Y denote a continuous random variable with finite mean  and finite
variance  2. Prove that for any   R+ \ {0},
P(|Y –  |   )   2/  2
Question 4:
Let Z denote a normal distribution with finite mean  and finite variance  2
(i.e. Z ~ N(  ,  2)). Prove that the mode of Z is  .
Question 5:
Consider the following set of population data:
{1, 3, 5, 6, 8, 10, 12, 13, 15, 16}
(a) Find the 35th percentile of the data set
(b) Calculate the mean, median and standard deviation of the data set
(c) Evaluate the range and the inter-quartile range of the data set
(d) Find the skewness and the excess kurtosis of the data set
(e) Suppose the data set is a set of sample data. Redo part (a) to part (d)
MA3518: Applied Statistics
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Question 6:
Consider the following sample of grouped data with class value X and class
frequency F:
X
F
10
3
12
5
14
7
16
12
18
6
20
3
22
1
(a) Find the sample mean of the grouped data
(b) Calculate the sample standard deviation of the grouped data
(c) Evaluate the sample skewness and excess kurtosis of the grouped data
~ End of Tutorial 1~