Lecture 2:
Review of statistics – one random variable
BUEC 333
Professor David Jacks
1
Many of the things we will be interested in this
class are random variables (RVs); that is,
variables whose outcome is subject to chance.
We do not know what value a RV will take until
we observe it.
Examples: outcome of coin toss;
value of the S&P/TSX in the future;
your starting salary after graduation.
Random variables
2
The actual value taken by a RV is an outcome.
We will use capital letters to denote a RV, and
lower case letters to denote particular outcomes.
Examples: Rolling two dice;
call the sum of the values rolled X;
a particular outcome might be x = 7.
Random variables
3
RVs are said to be discrete if they can only take
on a finite (that is, countable) set of values.
Examples:
1.) Outcome of a coin toss: {Heads, Tails}
2.) Outcome of rolling a die:
{1, 2, 3, 4, 5, 6}
3.) Handedness of next person you meet:
{Left, Right}
Discrete versus continuous RVs
4
RVs are said to be continuous if they can take on
a continuum (that is, uncountable infinite) set of
values.
Examples:
1.) Value of the S&P/TSX one year from
today: any positive number is
possible.
2.) Your starting salary after graduation:
any positive(?) number is possible.
Discrete versus continuous RVs
5
Associated with every possible outcome of a RV
is a probability which tells us how likely a
particular outcome is.
Pr(X = x) denotes the probability that the random
variable X takes the value x.
Pr(X = x) as the proportion of times that x occurs
in the ―long run‖ (in many repeated trials);
Probability
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1.) Probabilities of individual outcomes lie
between 0 and 1.
a.) If Pr(X = x) = 0, then outcome X = x
never occurs.
b.) If Pr(X = x) = 1, then outcome X = x
always occurs.
2.) The sum of the probabilities of all possible
individual outcomes always equals 1.
Properties of probability
7
Every RV has a probability distribution.
A probability distribution describes the set of all
possible outcomes of a RV, and the probabilities
associated with each possible outcome.
This is summarized by a probability distribution
function (or pdf).
Probability distributions
8
Example 1: tossing a fair coin
Pr (X = Heads) = Pr (X = Tails) = 1/2
Example 2: Rolling a die
Pr (X = 1) = Pr (X = 2) = …= Pr (X = 6)= 1/6
Example 3: # of times a laptop crashes before MT
Pr (X = 0) = 0.80, Pr (X = 1) = 0.10
Pr (X = 2) = 0.06, Pr (X = 3) = 0.03
Probability distributions
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An alternate way to describe a probability
distribution is the cumulative distribution
function (or cdf).
It gives the probability that a RV takes a value less
than or equal to a given value, Pr(X ≤ x).
Example: number of times a laptop crashes before
a midterm; the cdf measures the probability that a
Cumulative probability distribution
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1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
1
2
3
4
0
Cumulative probability distribution
1
2
3
4
11
Because continuous RVs can take an infinite
number of values, the pdf and cdf cannot
enumerate the probabilities of each of them.
Instead we describe the pdf and cdf using
functions.
Usual notation for the pdf is f(x).
Usual notation for the cdf is F(x) = Pr(X ≤ x).
The case of continuous RVs
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Because pdfs and cdfs of continuous RVs can also
be complicated functions, we will just use pictures
to represent them.
We plot outcomes, x, on the horizontal axis, and
probabilities, f(x) or F(x), on the vertical axis.
The cdf is an increasing function that ranges from
zero to one.
Graphing pdfs and cdfs of continuous RVs
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Remember: the area under the pdf gives the
probability that X lies in a particular interval.
Therefore, the total area under the pdf must be
equal to one; e.g., normally distributed test scores
1.00
0.20
cdf
0.80
pdf
0.15
0.60
0.10
0.40
0.05
0.20
0.00
0.00
0.0
25.0
50.0
75.0
100.0
0.0
25.0
50.0
Graphing pdfs and cdfs of continuous RVs
75.0
100.0
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The pdf and cdf effectively tell us ―everything‖ we
might want to know about a RV.
But sometimes we only want to describe particular
features of a probability distribution.
One feature of interest in this course is the
expected value or mean of a RV.
Another useful measure of the dispersion
Describing RVs
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Think of the expected value of a RV as its long run
average over many repeated trials.
More intuitively, can be thought of as the
―middle‖ of a probability distribution or a ―good
guess‖ of the value of a RV.
More precisely, it is a probability-weighted
average of all possible outcomes of X.
Expected values
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Example: laptop crashes before a midterm
f(0) = 0.80, f(1) = 0.10, f(2) = 0.06 ,
f(3) = 0.03, f(4) = 0.01
E(X) = 0 * (0.80) + 1 * (0.10) + 2 * (0.06)
+ 3 * (0.03) + 4 * (0.01)
Expected values
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The general case for a discrete RV when X can
take k values x1, x2,…, xk with associated
probabilities p1, p2,…, pk:
k
E ( X )   pi xi
i 1
Reminder about sigma notation: sigma represents
Even more exciting facts about E(X)
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We can think of E(X) as a mathematical operation
just like (+, -, *, or /).
Conveniently, it is also a linear operator which
means we can pass it through addition and
subtraction operators.
That is, if a and b are constants and X is a RV, then
E(a + bX) = a + E(bX) = a + bE(X)
Example where a = 5, b = 10, and
Even more exciting facts about E(X)
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Variance measures dispersion—how ―spread out‖
a probability distribution is.
A large (small) variance means a RV is likely to
take a wide (narrow) range of values.
Formally, if X takes one of k possible values x1,
x2,…, xk with associated probabilities p1, p2,…, pk:
2

Var ( X )  E ( X   x ) 
Variance
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Because Var(X) is measured in the square of the
scale of X, we often prefer the standard deviation:
 X  Var ( X )
which is measured on the same scale as X.
Example: variance of laptop crashes
Var(X) = (0 – 0.35)2 * (0.80) + (1 – 0.35)2 * (0.10)
+ (2 – 0.35)2 * (0.06) + (3 – 0.35)2* (0.03)
Variance and standard deviation
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In some sense, a RV’s probability distribution,
expected value, and variance are abstract
concepts.
More precisely, they are population parameters,
characteristics of the whole set of possible
observations of a RV.
An important aside and a preview of things to come
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As econometricians, our goal is to estimate these
parameters (with varying degrees of precision).
We do that by computing statistics from a sample
of data drawn from the population.
The usefulness of econometrics comes from
learning about
An important aside and a preview of things to come
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