Tutorial: Problem Set 1
Exercise 1 For two events A and B, the De Morgans laws imply that
(A B) = A B
and
(A B) = A B.
1. Use Venn diagrams to verify this equation
2. Prove this equation [hint: P (A B) = 1 P (A B)]
Exercise 2 Show that if one event A is c
Hypothesis Testing
Exercise 1 For which of the given p-values would the null hypothesis be rejected when
performing a level 0.05 test? a) 0.001 b) 0.021 c) 0.078 d) 0.047 e) 0.148
Exercise 2 A researcher wishes to test if the mean fat content of a type of
Point estimation
Exercise 1 Consider the following sample of observations on coating thickness for lowviscosity paint:
0.83
1.48
0.88
1.49
0.88
1.59
1.04
1.62
1.09
1.65
1.12
1.71
1.29
1.76
1.31
1.83
Assume that the distribution of coating thickness is nor
Introduction to Probability
Exercise 1 A ball is chosen (at random) from a bag containing 3 red and 4 blue balls.
What is the probability that the chosen ball is blue?
Exercise 2 A card is selected from a well shuffled pack of 52 cards (the experiment).
1
Discrete Random Variables
Exercise 1 1. 0.013; 2. 0.162; 3. 0.485
Exercise 2 1. 1/32;
2. X B(8, 1/32) and P (X > 2) = 1 P (X = 0) P (X = 1) P (X = 2) = 0.015.
Exercise 3
(133)(132)(132)(131)
= 0.030.
(528)
Exercise 4 see correction in the class website
Ex
Tutorial 4: Chi-squared goodness-of-fit test
1
Pearsons theorem
Let us consider r boxes B1 , . . . , Br and throw n balls X1 , . . . , Xn into these boxes independently of each other with probabilities
P (Xi B1 ) = p1 , . . . , P (Xi Br ) = pr
so that
p1
Tutorial: Problem Set 2
Exercise 1 Let X Exp() and let the random variable Y defined by Y = X 2 .
Find the cdf of X.
Find the 25th percentile of the distribution of X.
Find E[Y ] [hint: Y is a defined as a function of X]?
Exercise 2 Let Z N (0, 1) and
Joint distribution and Random Sample
Exercise 1 Given the values of the joint probability distribution of X and Y shown in
the table
X
-1
-1 1/8
1 1/2
Y
0
0
1/4
1
1/8
0
1. What is the marginal distribution of X?
2. What is the marginal distribution of Y ?
Tutorial: Problem Set 3
Exercise 1 Prove the following:
1. If 1 is an unbiased estimator for , and W is a zero mean random variable, then
2 = 1 + W is also an unbiased estimator for .
2. If 1 is an estimator for such that E[1 ] = a + b, where a 6= 0, show
Hypothesis Testing
Exercise 1 We reject H0 if and only if pvalue = 0.05. a) Reject H0 b) Reject H0
c) Do not reject H0 d) Reject H0 e) Do not reject H0
Exercise 2 This is an upper-tailed test. Let be the mean fat content of the hotdogs.
Then we are intere
Confidence Intervals
Exercise 1 A sample of size 50 is drawn from a population with = 10. The sample
mean is 20.26. Calculate and interpret a 95% confidence interval for .
Exercise 2 A population is normally distributed with = 5.6, but is unknown. A
rando
Inference for Two Samples Simple Linear Regression
Exercise 1
sizes.
Y ) = E(X)
E(Y ) = 4.1 4.5 = 0.4, irrespective of sample
1. E(X
Y ) = V ar(X)
+ V ar(Y ) =
2. V ar(X
0.2691.
1.82
100
2
2
+ 100
= 0.0724. The standard deviation is
3. A normal curve
Discrete Random Variables
Exercise 1 A fair die is rolled 10 times.
1. What is the probability that it shows the number 6 exactly 5 times?
2. What is the probability that it does not show the number 1 at all?
3. What is the probability that it shows the n
Continuous Random Variables
Exercise 1 Determine whether the following functions are pdf
3
1
4 x 16
(x 1)x 0 x 2
4 x
2
f (x) =
g(x) =
0 otherwise
0 otherwise
ln 2
1
2
x
2x
2
x(ln x)
x(ln x)2
v(x) =
u(x) =
0 otherwise
0 otherwise
Exercise 2 The pdf of a
Inference for Two Samples Simple Linear Regression
Exercise 1 The average lifetimes of Duracell Alkaline AA batteries and Eveready Energizer Alkaline AA batteries were given as 4.1 hours and 4.5 hours respectively. Suppose
these are the population average
Point estimation
Exercise 1
1. We use the sample mean, x =
P
xi
n
= 1.3481.
2. Because
we assume normality, the mean = median, so we also use the sample mean
P
x = nxi = 1.3481.
3. We use the 90th percentile of the sample:
+ 1.28
= x + 1.28s = 1.3481 +
Tutorial: Problem Set 1
Exercise 1
P (A B) = 1 P (A B)
= 1 P (A) P (B) + P (A B)
= P (A) P (B A)
= P (A B)
Exercise 2
A B P (B A) = 0
Hence P (A) = P (AB) and P (B) = P (AB)+P (AB). So P (B) = P (A)+P (AB),
since 0 P (A B) 1, we have P (A) P (B).
Exercise
Continuous Random Variables
Exercise 1 Only g and v are pdf.
Exercise 2 1; c = 3/8; 2. for 0 x 2 F (x) = x3 /8 and P (1 < X < 1.5) = 19/64; 3.
E[X] = 3/2 V ar(X) = 3/20
Exercise 3 P (X 2.6) = 0.926 and P (1 < X < 4) = 0.350. If 0 < x f (x) = ex and
f (x)