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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 Page 2 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 Page 3 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~