STA2002: Probability and Statistics II Fall 2024
Homework #1
This is the homework for week 2. The due date is 11:59 p.m., September 22. Late submission will
automatically result in “0” grade. Please include codes for the program part in your submission.
1 Computation Questions
(Question1): (30 points) i. Suppose random variable X satisfies Possion distribution with param-
eter λ > 0, i.e., the probability mass function
λk
P rob(X = k) = exp(−λ), where k = 0, 1, · · · ,
k!
Calculate EX, V ar(X), and its moment generating function H(t) = E exp(tX), where t ∈ R.
λ
ii. Denote F (a) to be the cumulative distribution function of X, prove that F (a) ≥ 1 − a
for any a > 0.
iii. Now suppose random variable Y satisfies uniform distribution on [0, 1], i.e., its density
f (t) = 1 on [0, 1]. Derive the mean EY , the variance V ar(Y ), the skewness γ, and the
Kurtosis Kurt(Y ).
iv. Suppose Z1 , Z2 , · · · , Zn are i.i.d. (independent and identical distributed) random vari-
ables, prove that
Xn n
X
V ar( Zi ) = V ar(Zi ).
i=1 i=1
Hint: you can use this fact when you prove this question: If Z1 , Z2 , · · · , Zn are independent,
then for any function h, h(Z1 , · · · , Zk ) and Zk+1 is also independent.
(Question2): (20 points)
(Hint, see problem 5.3–8. in the textbook) Suppose two independent claims X and Y are
made on two insured homes, where each claim has probability density function
4
f (x) = , 1 < x.
x5
Find the expected value and the variance of the larger claim max(X, Y ).
(Question3): (30 points)
Suppose X and Y are independent random variables with mean EX = EY = µ ̸= 0 and
variance V ar(X) = V ar(Y ) = σ 2 > 0. Find the correlation coefficient
Corr(XY, Y ).
2 Programming Questions
(20 points)
Implement the following experiment using Jupyter or R:
1
i. Generate 500 i.i.d. random variables X1 , · · · , X500 with uniform distribution on [0, 1].
ii. Calculate and record the sample mean, sample variace, sample skewness, and sample Kurtosis
(See the “Estimator” column of Table 1 of the lecture notes “probability review”).
iii. Repeat experiment for 300 times, plot four histograms of the sample values. Make comparisons
between the sample values and the true values.
