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EE 503: Homework 3
Due: 09/23/2021
Student’s name:
Instructor:
1
1) An urn contains n + m balls, of which n are red and m are black. They are
withdrawn from the urn, one at a time and without replacement. Let X is the
number
of
red
balls
removed
before
the
first
black
is
chosen.
We
are
interested in determining E[X]. To obtain this quantity, number the red balls
from 1 to n. Now define the random variables Xi, i=1,...,n, by
1,
Xi =
0,
if red ball i is taken before any black ball is chosen
Otherwise
a) Express X in terms of Xi.
ANS:
The random variable X represents the number of red balls removed before the first black ball is chosen.
Thus, X can be represented in terms of Xi, as follows:
X = ∑𝑛𝑖=𝑖 𝑋𝑖
= X1 + X2 + …+ Xn
b) Find E[X].
ANS:
If 1 red ball is chosen before black ball then,
𝑛 𝑛
X1 = ( )(𝑚+𝑛)
1
If 2 red ball is chosen before black ball then,
𝑛 𝑛
𝑛 − 1 𝑛−1
X2 = ( )(𝑚+𝑛) + (
)(𝑚+𝑛−1)
1
1
If n red ball is chosen before black ball then,
𝑛 𝑛
𝑛 − 1 𝑛−1
1 −1
Xn = ( )(𝑚+𝑛) + (
)(𝑚+𝑛−1)+…+ ( )(𝑚+1)
1
1
1
𝑛
E[X]= E [∑𝑛𝑖=𝑖 𝑋𝑖 ]= E [X1 + X2 + …+ Xn] =E [X1]+ E [X2]+…+ E [Xn]= 𝑚+1
E[X]=
𝑛
𝑚+1
2
2) Let X be a uniformly distributed random variable in the interval [−π,π]. What is the cdf of Y
=tanX?
ANS:
𝑙𝑒𝑡 𝑥 ~𝑣(−𝜋, 𝜋)
𝑝𝑑𝑓 𝑜𝑓 𝑥;
𝑓(𝑥) =
1
, ...
