Now Suppose That the Urn Initially Contains 20 Red 10 Blue and 8 Green Balls
MTHE/STAT 353 – Winter 2015
Homework Assignment 1
Assignment 1 — due Friday, Jan. 16
1. An urn contains 10 balls, i of which are of colour i , for i = 1, 2, 3, 4, and 7 balls are
drawn at random without replacement. Let Xi denote the number of balls in the
sample that are of colour i.
(a) Find P (X1 = 1, X2 = 1, X3 = 2, X4 = 3).
(b) Find the marginal probability mass function of X3 .
(c) Find P (X4 ≤ 2).
2. Suppose an urn initially contains one red ball, one blue ball, and one green ball. At
each draw, a ball is randomly selected from the urn, replaced, and an additional ball
of the same colour as the drawn ball is added to the urn. Thus, after n draws there
are n + 3 balls in the urn. After n draws, let X be the number of times a red ball
was drawn, Y the number of times a blue ball was drawn, and Z the number of times
a green ball was drawn. Compute the joint probability mass function of the random
vector ( X, Y, Z ).
3. Let X1 be a discrete random variable and let X2 be a continuous random variable. The
joint pmf/pdf of (X1 , X2 ) is a function f :R2 → [0, ∞ ) satisfying
P( X1 = x1 , X2 ∈ A) = Z
{x2 ∈A }
f( x1 , x2 ) dx2.
for all x1 ∈R and all A⊂R .
(a) Give expressions for the marginal pmf of X1 and the marginal pdf of X2 in terms
of f.
(b) Suppose that (X1 , X2 ) have joint pmf/pdf
f( x1 , x2 ) = 9!
29 x1 !(10−x1 )! for x 1 = 1,..., 10 and 0 < x 2 < x 1
0 otherwise.
Find the marginal pmf of X1 . Are X1 and X2 independent?
(c) Find P (X2 > 8), where (X1 , X2 ) have the joint pmf/pdf from part(b).
4. Suppose that X1 , . . . , Xn are jointly continuous with joint probability density function
f( x1 , . . . , xn ) = 1
2π n/2
exp − 1
2h x 2
n+
n−1
X
i=1
(xi −xn )2 i .
Find the joint marginal pdf of ( X1 , . . . , Xn−1 ). Are X1 , . . . , Xn−1 mutually indepen-
dent?
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Source: https://www.studocu.com/en-ca/document/queens-university/probability-ii/probability-ii-assignments-hw-1-6-2015/204302
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