Statistics 630 Assignment 3: Chapter 2 Problems and Solutions

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Statistics 630 - Assignment 3
(due Thursday, February 7, 2019, 8am CST)
Name ______________________________________________________________
Email Address ______________________________________________________
Chapter 2 Problems
2.4.2
W Uniform[1, 4].∼
(a)
(b)
(d)
2.4.3
Z Exponential(4)∼ ; where
(a)
(c)
2.4.4
(a) f (x) = cx on (0, 1) and 0 otherwise.

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(c) f (x) = cx1/2 on (0, 2) and 0 otherwise
2.4.7
Let M > 0, and suppose f (x) = cx2 for 0 < x < M, otherwise f (x) = 0. For
what value of c (depending on M) is f a density?
Solution:
2.4.19
(Weibull(α) distribution) Consider, for α > 0 fixed, the function given by f (x) = α (1 + x)
−α−1 for 0 < x < ∞ and 0 otherwise. Prove that f is a density function.
Solution:
Therefore, f (x) = α (1 + x) −α−1 is a density function
2.4.22
(Laplace distribution) Consider the function given by f (x) = e−|x| /2 for −∞ < x < ∞ and 0
otherwise. Prove that f is a density function.
Solution:
We know that for a valid density function,

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Therefore, f is a density function
2.5.3
a) ; This is a valid cumulative distribution function because it is non
decreasing, right continuous, and
c) . This is a valid cumulative distribution function because it is non
decreasing, right continuous, and
d) . This is not a valid cumulative distribution function because
f) . This is not a valid cumulative distribution function because it
does not approach 0 as
g) . This is not a valid cumulative distribution function because it
does not approach 0 as
2.5.5
Let Y N(−8, 4)∼