SAMPLE STATISTICS
A random sample x1 , x2 , . . . , xn from a distribution f (x) is a set of independently and identically variables with xi f (x) for all i. Their joint p.d.f is
n
f (x1 , x2 , . . . , xn ) = f (x1 )f (x2 ) f (x2 ) =
f (xi ).
i=1
The samp
EXERCISES IN STATISTICS
Series A, No. 7
1. Let x and y be jointly distributed random variables with conditional expectations which can be written as E (y |x) = + x and E (x|y ) = + y .
Express and in terms of the moments of the joint distributions and
sho
EXERCISES IN STATISTICS
Series A, No. 6
1. Let x1 and x2 have the joint p.d.f f (x1 , x2 ) = x1 + x2 with 0 x1 , x2 1.
Find the conditional mean of x2 given x1 .
2. Find the P (x < y |x < 2y ) when f (x, y ) = e(x+y) . Draw a diagram to
represent the even
EXERCISES IN STATISTICS
Series A, No. 5
1. Find the moment generating function of x f (x) = 1, where 0 < x < 1,
1
and thereby conrm that E (x) = 1 and V (x) = 12 .
2
2. Find the moment generating function of x f (x) = aeax ; x 0.
3. Prove that x f (x) = x
EXERCISES IN STATISTICS
Series A, No. 3
1. If f (x) = a + bx2 ;
2. Let f (x) = 1;
0 x 1, determine a and b such that E (x) = 3 .
4
0 x 1. Find
(a) the mean and variance of x,
(b) the mean and variance of x2 .
3. The probability that x buses will pass me b
EXERCISES IN STATISTICS
Series A, No. 2
1. Let A1 , A2 be subsets of a sample space S . Show that
P (A1 A2 ) P (A1 ) P (A1 A2 ) P (A1 ) + P (A2 ).
2. Find the probabilities P (A), P (B ) when A, B are statistically independent
events such that P (B ) = 2P
EXERCISES IN STATISTICS
Series A, No. 1
1. The number of children recorded to each of 25 families were
2, 4, 1, 0, 1, 3, 0, 4, 2, 6, 0, 0, 2, 3, 1, 5, 4, 0, 3, 1, 2, 5, 3, 4, 1.
(a) Construct a frequency table and a graph for the distribution of family
si
EC2019 SAMPLING AND INFERENCE, 2010
QUESTIONS IN PREPRATION FOR THE EXAMINATION
The examination asks for THREE answers
The time allowed is TWO hours
1. Give an account of the axioms of Boolean algebra and, in the process,
compare them with the axioms of a
THE POISSON DISTRIBUTION
The Poisson distribution is a limiting case of the binomial distribution which
arises when the number of trials n increases indenitely whilst the product
= np, which is the expected value of the number of successes from the trial
EXERCISES IN STATISTICS
Exercise M01
1. The probability density function governing the minutes of time t spent
waiting outside a telephone box is given by f (t) = aeat .
(a) Determine the probability of having to wait for more than t minutes.
(b) Show tha
EXERCISES IN STATISTICS
Series A, No. 8
1. The value of the mean of a random sample of size 20 from a normal population is x = 81.2 Find the 95% condence interval for the mean of the
population on the assumption that the variance is V (x) = 80.
2. Let x b
EXERCISES IN STATISTICS
Series A, No. 9
1. The average length of a nger bone of 10 fossil skeletons of the proconsul
hominid is 3.73cm, and the standard deviation is 0.34cm. Find 80% and 90%
condence intervals for the mean length of the bone in the specie
MULTIVARIATE DISTRIBUTIONS
Each element a random vector x = [x1 , x2 , . . . , xn ]0 describes an aspect of a statistical outcome. We
write x Rn to signify that x is a point in a real n-space.
A function f (x) = f (x1 , x2 , . . . , xn ) assigning a proba
BIVARIATE DISTRIBUTIONS
The probabilities f (xi ) of the values cfw_x1 , x2 , . . . , xn assumed by a discrete random variable x are such
that
f (xi ) 0 for all i
and
f (xi ) = 1.
i
x
1
1
For example:
f (x)
0.25
0.75
If x and y take values cfw_x1 , x2 ,
MOMENT GENERATING FUNCTIONS
The natural number e. The number e = cfw_2.7183 . . . is dened by
n
1
e = lim(n ) 1 +
.
n
The binomial expansion indicates that
(a + b)n = an + nan1 b +
n(n 1) n2 2 n(n 1)(n 2) n3 2
a
b+
a
b + .
2!
3!
Using this, we get
n
2
3
PROBABILITY DISTRIBUTIONS: (continued)
The change of variables technique. Let x f (x) and let y = y (x) be a monotonic
transformation of x such that x = x(y ) exists.
Let A be an event dened in terms of x, and let B be the equivalent event dened in terms
DISCRETE AND CONTINUOUS
PROBABILITY DISTRIBUTIONS
Probability mass functions
If x cfw_x1 , x2 , x3 , . . . is discrete, then a function f (xi ) giving the probability that x = xi is called a
probability mass function. Such a function must have the propert
AXIOMATIC PROBABILITY AND POINT SETS
The axioms of Kolmogorov. Let S denote an event set with a probability measure P
dened over it, such that probability of any event A S is given by P (A). Then, the
probability measure obeys the following axioms:
(1) P
ELEMENTARY PROBABILITY
Summary measures of a statistical experiment. Let us toss the die 30 times and let
us record the value assumed by the random variable at each toss:
1, 2, 5, 3, . . . , 4, 6, 2, 1.
To summarise this information, we may construct a fr
EXERCISES IN STATISTICS
Exercise M01
1. The probability density function governing the minutes of time t spent
waiting outside a telephone box is given by f (t) = aeat .
(a) Determine the probability of having to wait for more than t minutes.
(b) Show tha
EXERCISES IN STATISTICS
Series A, No. 10
1. A horticulturist considers that a batch of seeds is worth sowing if 50%
of the resulting owers are going to be pure white. To test the worth of
a particular batch, he sows eight seeds with the intention of sowin
HYPOTHESIS TESTING
We have already described how the sample information is used to estimate the
parameters of the underlying population distributions. Now, we intend to use our
sample information to test existing presumptions regarding the values of param
THE THEORY OF POINT ESTIMATION
A point estimator uses the information available in a sample to obtain a single
number that estimates a population parameter. There can be a variety of estimators
of the same parameter that derive from dierent principles of
EXERCISES IN STATISTICS
Series A, No. 10
1. A horticulturist considers that a batch of seeds is worth sowing if 50% of the
resulting owers are going to be pure white. To test the worth of a particular
batch, he sows eight seeds with the intention of sowin
EXERCISES IN STATISTICS
Series A, No. 9
1. The average length of a nger bone of 10 fossil skeletons of the proconsul
hominid is 3.73cm, and the standard deviation is 0.34cm. Find 80% and 90%
condence intervals for the mean length of the bone in the specie
EXERCISES IN STATISTICS
Series A, No. 8
1. The value of the mean of a random sample of size 20 from a normal population is x = 81.2 Find the 95% condence interval for the mean of the
population on the assumption that the variance is V (x) = 80.
Answer. We
EXERCISES IN STATISTICS
Series A, No. 7
1. Let x and y be jointly distributed random variables with conditional expectations which can be written as E (y |x) = + x and E (x|y ) = + y .
Express and in terms of the moments of the joint distributions and
sho
EXERCISES IN STATISTICS
Series A, No. 6
1. Let x1 and x2 have the joint p.d.f f (x1 , x2 ) = x1 + x2 with 0 x1 , x2 1.
Find the conditional mean of x2 given x1 .
Answer. First let us show that
f (x1 , x2 )dx2 dx1 = 1 :
x2
x1
We have
(x1 + x2 )dx1 dx2 =
x2
EXERCISES IN STATISTICS
Series A, No. 5
1. Find the moment generating function of x f (x) = 1, where 0 < x < 1,
1
and thereby conrm that E (x) = 1 and V (x) = 12 .
2
Answer: The moment generating function is
1
xt
ext dx
M (x, t) = E (e ) =
0
t
xt 1
=
But
EXERCISES IN STATISTICS
Series A, No. 4
1. Ten jurors are selected at random from a long list of names of which one
third belong to women. The selection is regarded as unsatisfactory if either
sex is represented by less than three people. Find the probabi