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MTH201: Probability and Statistics
Quiz 2 Answers
Date: February 4, 2015
1. Given below is a cumulative density function F(X) for random variable X, which can only take discrete values
from the set {1,2,3,4,5,6,7} :
x
x≤0
x≤1
x≤2
x≤3
x≤4
x≤5
x≤6
x≤7
F(x)
0
0.038
0.308
0.346
0.423
0.577
0.808
1
Find the probability mass function for X. (5 marks)
Sol:
x
x=0
x=1
x=2
x=3
x=4
x=5
x=6
x=7
f(x)
0
0.038
0.27
0.038
0.077
0.154
0.231
0.192
2. Given below is joint probability distribution for discrete random variables X and Y. Both the variables can only
take discrete values from the set {10,20,30} :
PXY(x, y)
x = 10
x = 20
x = 30
1
1
4
12
1
1
y = 20
6
18
1
1
y = 30
36
12
Are random variables X and Y independent? Justify. (5 marks)
Sol: P(x=10) = ½; P(x=20) = 1/6; P(x=30) = 1/3;
P(y=10) = ½; P(y=20) = 1/3; P(y=30) = 1/6;
y = 10
1
6
1
9
1
18
Show P(X=x1, Y=y1) = P(X=x1)*P(Y=y1) for all possible 9 pairs
3. The probability density functions of independent random variables X and Y are given are follows:
1 2
2<𝑥≤3
𝑓(𝑥) = {19 𝑥 ,
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
3
3
0<𝑦≤1
𝑓(𝑦) = {𝑦 + 4 ,
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Compute Var (3X + Y + 11). (5 marks)
Sol: 13.406
(1.5 marks)
4. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice,
and if tails, then one-half of the value that appears on the die. Determine her expected winnings. (5 marks)
Sol: 4.375
1