Slide set 7
Stat 330 (Spring 2015)
Last update: January 16, 2015
Stat 330 (Spring 2015): slide set 7
Random Variables
Intuitive idea: If the value of a numerical variable depends on the outcome
of an experiment, we call the variable a random variable.
Random Variable A function X : Ω 7→ R is called a random variable.
Example: Very simple Dartboard
Imagine we throw three darts on this board one by one and we are interested
in the number of times the red area has been hit. This count is a random
variable!
Also note that P (red) =
1
9
on any throw.
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Stat 330 (Spring 2015): slide set 7
More formally:
Dartboard (continued...)
We define X to be the function that assigns the number of times that the
red area is hit in a sequence of three throws.
X(ω) = k, if the outcome ω has k hits to the red area, and 3 − k hits to
the gray area.
X(ω) is then an integer between 0 and 3 for every possible sequence of
throws of 3 darts.
That is, the set of possible values for X(ω) is {0, 1, 2, 3}
Notation: To avoid cumbersome notation, we write
{X = x}
for the event
{ω|ω ∈ Ω and X(ω) = x}.
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Stat 330 (Spring 2015): slide set 7
Practice with notation
Example Suppose, 8 bits are sent through a communication channel. Each
bit has a certain probability to be received incorrectly. We are interested in
the number of bits that are received incorrectly.
Use random variable X to “count” the number of wrong bits received. X
assigns a value between 0 and 8 to each sequence in the sample space.
That is, the possible values for X are {0, 1, 2, 3, 4, 5, 6, 7, 8}
Examples of events and equivalent expressions using X.
• No wrong bits received: {X = 0}
• At least one wrong bit received: {X ≥ 1}
• Exactly 2 wrong bits received: {X = 2}
• Between 2 and 7 wrong bits (inclusive) received: {2 ≤ X ≤ 7}
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Stat 330 (Spring 2015): slide set 7
Image of R.V.: all possible values X can take
Definition:
The image of a random variable X
Im(X) := {x : x = X(ω) for some ω ∈ Ω}.
is defined as
1. Put a disk drive into service, measure Y = “time till the first major
failure”.
Im(Y ) = (0, ∞).
image of Y is an interval (uncountable image) → Y is a continuous
random variable.
2. Communication channel: X = “# of incorrectly received bits”
Im(X) = {0, 1, 2, 3, 4, 5, 6, 7, 8} is a finite set → X is a discrete random
variable.
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Stat 330 (Spring 2015): slide set 7
Discrete R.V.s
• Assume X is a discrete random variable. The image of X is therefore
countable and can be written as {x1, x2, x3, . . .}
• Very often we are interested in probabilities of the form P (X = x). We
can think of this expression as a function, that yields different probabilities
depending on the value of x.
Example: Dartboard What is the probability that you hit the red square
exactly twice in 3 throws?
The event “exactly 2 reds” is formally written as {ω : X(ω) = 2}, and with
the simpler notation for this event is X = 2.
P (2 reds) = P (X = 2)
= P (RRG or RGR or GRR)
= P (RRG) + P (RGR) + P (GRR)
=
1
9
· 19 · 98 + 19 · 89 · 19 + 89 · 19 · 19 =?
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Stat 330 (Spring 2015): slide set 7
Probability Mass Function
Definition: Probability Mass Function, PMF
The function pX (x) := P (X = x) is called the probability mass function
of X.
Properties of PMF:
pX is the pmf of X, if and only if
(i) 0 ≤ pX (x) ≤ 1 for all x ∈ {x1, x2, x3, . . .} (all values must be between
0 and 1)
(ii)
P
i pX (xi )
= 1 (the sum of all values is 1)
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Stat 330 (Spring 2015): slide set 7
Examples:
These properties give us an easy method to check, whether a function is a
probability mass function
Experiment: Which of the following functions is a valid probability mass
function?
1.
x
pX (x)
-3
0.1
-1
0.45
0
0.15
5
0.25
7
0.05
2.
y
pY (y)
-1
0.1
0
0.45
1.5
0.25
3
-0.05
4.5
0.25
3.
z
pZ (z)
0
0.22
5
0.17
7
0.18
1
0.18
3
0.24
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Stat 330 (Spring 2015): slide set 7
More Examples of Discrete R.V.’s
Example: Roll of a fair die
Let Y be the number of spots on the upturned face of a die.
Obviously, Y is a random variable with image Im(Y ) = {1, 2, 3, 4, 5, 6}.
Assuming, that the die is a fair die means, that the outcomes of each face
turning up are equally likely i.e., probabilities of each face turning up are
equal.
The probability mass function for Y therefore is
1 2 3 4 5 6
x
pX (x) 16 16 16 16 16 16
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Stat 330 (Spring 2015): slide set 7
More Examples of Discrete R.V.’s
Example: Roll of a doctored die
The diagram shows all six faces of a particular die.
If Z denotes the number of spots on the upturned
face after toss this die, what is the probability mass
function for Z?
Assuming, that each face of the die appears with
the same probability, we have 1 possibility to get
a 1 or a 4, and two possibilities for a 2 or 3 to
appear, which gives a probability mass function
for Z as:
z
p(z)
1
2
3
4
1/6 1/3 1/3 1/6
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Stat 330 (Spring 2015): slide set 7
Statistics of R.V.s
Gamblers Luck!: Toss a die. Let X

 1, 3 or 5
2 or 4
X=

6
be the number of spots turned up, then if
I pay you $X
you pay me $2X
no money changes hands.
What amount of money do I expect to win?
For that, we look at another function, h(x), that counts the money I win
with respect to the number of spots:

 −x
2x
h(x) =

0
for x = 1, 3, 5
for x = 2, 4
for x = 6.
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Stat 330 (Spring 2015): slide set 7
Gambling Example Continued...
In
In
In
In
In
In
1/6
1/6
1/6
1/6
1/6
1/6
of
of
of
of
of
of
all
all
all
all
all
all
tosses
tosses
tosses
tosses
tosses
tosses
X
X
X
X
X
X
will
will
will
will
will
will
be
be
be
be
be
be
1,
2,
3,
4,
5,
6,
and
and
and
and
and
and
I
I
I
I
I
I
will
will
will
will
will
will
gain
gain
gain
gain
gain
gain
-1
4
-3
8
-5
0
dollars
dollars
dollars
dollars
dollars
dollars
In total I expect to get
1
6
· (−1) + 16 · 4 + 61 · (−3) + 16 · 8 + 16 · (−5) + 61 · 0 =
3
6
= 0.5
dollars per play.
In this example, we are calculating the expected value of the function h(X)
We denote this by E(h(X)) and define this mathematically in the next
lecture.
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