The bivariate normal distribution
We say that (X, Y ) have bivariate normal distribution if their joint density is
fX,Y (x, y)
2 X Y
2 1 2
2 (x X ) (y Y )
where X , Y , X , Y and are constants.
Independent variable: The
variable you manipulate or are
Dependent variable: The
variable that you are measuring
X axis: The horizontal axis. (abscissa)
Y axis: The vertical axis. (Ordinate)
z-SCORES ANDTHE NORMAL
A z-score is a location on the
distribution. A z-score also
automatically communicates the raw
scores distance from the mean
A z-score describes a raw scores
location in terms of how far above or
below the mean
More General Annuities
Payment period different from interest period
Payment period is longer than interest conversion period
Given that the effective rate of interest per month is 1%.
payment period = 2 months
Time Value of Money
Time value of money means that 1 received today is not equivalent to (usually worth
more than) 1 received one year in the future. The time value of money depends critically
on how interest is calculated.
Interest is the
Amortisation Schedules and Sinking Fund
Two commonly used methods of repaying a loan:
The amortisation method
The borrower repays the lender by means of instalment payments at periodic intervals.
The sinking fund method
In each period the b
A random variable is a numerical outcome of a
statistical experiment, i.e., a rule that assigns
a number X() to each outcome in .
eg X = number of students who will come to
the next lecture
We will consider two kinds of random variable:
Often, interested in more than one rv at a
eg For a randomly selected person,
X = height, Y = weight.
eg For a sample from a biological population,
X = number of predators, Y = number of prey.
So far, we only know how to d
Continuous random variables
Often we deal with random variables which
may take a continuum of values, rather than
a discrete set of possible values. For example,
If X is a continuous rv, its (cumulative) distribution function,
F (x) = Pr(X
Sum of independent Normal rvs
Recall that if X N (0, 1), then MX (t) = et /2.
In general, if X N (, 2), then MX (t) =
Now suppose Xi N (i, i2), i = 1, , n with
Xi mutually independent. The random variable
V = X1 + X2 + + Xn has MGF given b
2, t, F distributions
Definition: The continuous random variable
U is said to have the chi-squared distribution
with 1 degree of freedom if U = Z 2 where
Z N (0, 1).
Notation: U 2
Density of 2
For 0 u < ,
P (U u) = P Z u
= P uZ u
= 2P 0 Z u
Probability generating function (PGF)
For a discrete rv X the Probability Generating
Function GX (t) is a function of t (, )
GX (t) = E tX
whenever the mean exists.
So if X takes values 0, 1, 2, . . . with probabilities
p(0), p(1), p(2), .
MATH264: Statistical Theory and Methods II
Dr Kai Liu
1. Events and Probabilities
Example 1 Math264 with total 78 students
registered in 2008.
A: At least 20 students would come to the
B: At least 25 students would come
C: At most 23 wou
Functions of a random variable
Suppose Y = g(X) for some continuous rv X
and function g. What is the distribution of Y ?
If X N (, 2) and Y = g(X) = aX + b for
numbers a, b, is it true that Y N (a +
b, a2 2)?
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