Government College University Geometric Distribution Statistics Questions

Homework assignment 4Due: Wednesday, June 29, 2022 11:59 pm (Pacific Daylight Time)
Q1 (10 points)
If X is an Exp(λ) random variable and a > 0 is a fixed number, let Y be a new random variable defined by
Y = aX
.
a) Write down the c.d.f. of X, and use it to find the c.d.f. of Y ;
b) What is the distribution of Y ? (This should be one of the distributions that you met in Chapter 3.)
Q2 (10 points)
If X is an Exp(λ) random variable, set Z = ⌈X⌉ (recall the ceiling function ⌈⋅⌉ means round up to the next integer,
so that ⌈4.231⌉ = 5 , ⌈2⌉ = 2 and ⌈−1.3⌉ = −1 ).
a) What is the range of values taken by Z ?
b) Find the p.m.f. of Z ;
c) Show that Z has a geometric distribution and identify the parameter.
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Q3 (10 points)
Suppose that X, Y , Z are discrete random variables taking values in the natural numbers with joint p.m.f.
fX,Y ,Z (i, j, k)
and joint c.d.f. FX,Y ,Z (i, j, k) (written F (i, j, k) for short).
Show that fX,Y ,Z (i, j, k) is equal to
F (i, j, k) − F (i − 1, j, k) − F (i, j − 1, k) − F (i, j, k − 1)
+ F (i, j − 1, k − 1) + F (i − 1, j, k − 1) + F (i − 1, j − 1, k)
− F (i − 1, j − 1, k − 1).
Q4 (10 points)
Let X be distributed as Bern( 12 ) and Y be distributed as Bern( 23 ).
a) Write down the p.m.f.s of X and Y , fX and fY .
b) Find all possible joint p.m.f.s of (X, Y ) with marginals fX and fY as above.
c) Amongst these joint p.m.f’s, find the joint p.m.f. that maximizes the probability that X and Y take the same value.
Q5 (10 points)
Suppose that X and Y are discrete random variables taking values in {1, 2, 3} with p.m.f.s
fX (1) = fX (2) = fX (3) =
1
3
and fY (1) = 18 , fY (2) = 12 and fY (3) = 38 .
Show that there is a possible joint distribution for (X, Y ) compatible with the above marginals so that Y ≥ X with
probability 1.
Q6 (10 points)
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If X is a standard normal random variable, what is E(|X|)?
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