Transcript PMF and Examples

PMF and Examples
PMF
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We have introduced the concept of PMF.
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1. It is short for “Probability Mass Function”.
2. It is used for discrete random variables.
3. It specifies the probability of each sample point
in the sample space of a random variable.
4. For each sample point, xi, in the sample space,
p(xi) is non-negative
5. The sum of all p(xi) should be 1.
PMF
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Form of PMF
X
x1
x2
x3
…
xn
P(X)
P(x1)
P(x2)
P(x3)
…
P(xn)
Example 1
Roll a 6-sided fair
die and let the
random variable X
be the outcomes of
rolling.
 Sample space:
{1, 2, 3, 4, 5, 6}
 PMF:
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X
P(X)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Example 2
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Roll a 6-sided fair
die once and let
random variable Y
be the outcome that
whether we get a
number of greater
than 2.
X
P(X)
1
(greater
than 2)
0
(less than
or equal
2)
4/6 or 2/3
2/6 or 1/3
Summary
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In order to find the PMF, we need to
know
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1. What experiment we are talking about
2. How the random variable is defined
3. Find the sample space and
corresponding probability
Example 3
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4 players are playing
a deck of 52 cards
and let X be the
number of aces one
player could have,
find the PMF of X.
X
0
1
2
3
4
P(X)
Example 4
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Homer is playing in a game which has two
parts. He must shoot at a target first. If he
hits the target, he will then be asked a 5
choice multiple choice question. If Homer hits
the target, he will be rewarded $20 and if he
gets the question, he will be rewarded $40.
Assuming that Homer can hit the target with
60% chance and has no clue about the
question, let X be the possible pay-out Homer
can get from the game, find the PMF of X.
Example 4
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1. Possible outcomes for Homer from the
game {make $60, make $20, get nothing}
2. Sample space {60, 20, 0}
3. P(Homer got 0)=P(Homer missed the
target)
4. P(Homer got $20)=P(Homer hit the target
but got the question wrong)
5. P(Homer got $60)=P(Homer hit the target
and got the question correctly)
Example 4
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Let A={Homer hit the target} and
B={Homer got the question correctly}
Then
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P(0)=P(Ac)=1-P(A)
P(20)=P(ABc)=P(Bc|A)P(A)
P(60)=P(AB)=P(B|A)P(A)
Example 4
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Finally, the PMF is
X
P(X)
0
0.4
20
0.48
60
0.12
Example 5
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Bart is playing with a fair coin. He
decides he will stop until he sees the
first head or three tails. Let X be the
number of tosses Bart will make and
find the PMF of X.
Example 5
Possible values of X: {1, 2, 3}
P{X=1}
P{X=2}
P{X=3}
Example 5
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PMF of X
X
P(X)
1
1/2
2
1/4
3
1/4
Find PMF on transforms of X
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Given the PMF of X,
what is the PMF of
2x+1?
X
P(X)
1
1/2
2
1/4
3
1/4
Find PMF on transforms of X
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Since there is a one-toone correspondence
between X and Y, if we
know X, we know Y
automatically. The
probability that X=xi is
exactly the same as the
probability that
Y=2xi+1
X
Y
P(Y)
1
3
1/2
2
5
1/4
3
7
1/4
Find PMF on transforms of X
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What if we are looking
for the PMF for Z=X^2?
Similarly, if we know X,
we know exactly what Z
is here, so we can reconstruct the PMF chart
in the form of
X
Z
P(Z)
1
1
1/2
2
4
1/4
3
9
1/4
Find PMF on transforms of X
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How about the PMF for
Z=X^2 if the PMF of X
is like the one on the
right?
X
P(X)
-2
1/2
1
1/4
2
1/4
Find PMF on transforms of X
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In this case, there are
two X values that will
give the same Z value,
(-2)^2=2^2=4.
Therefore, we will need
to merge some of the
probabilities to create
the PMF chart
The PMF should look a
little different.
Z
P(Z)
1
1/4
4
3/4
Example 6
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The PMF of X is given
and we want to find the
PMF of Z=X^2
First, we want to verify
it is a valid PMF, add up
all X’s to check whether
it is 1.
X
P(X)
-2
-1.5
1/8
1/16
-1
1/8
0
1/4
1
3/8
2
1/32
3
1/32
Example 6
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Z=X^2, therefore, we
should have a different
set of values for Z and
we want to keep track
of the probabilities too.
For example, Z=1 if X=1 with p=1/8 and X=1
with p=3/8; Z=2.25 if
X=-1.5 with p=1/16,
etc.
Z
P(Z)
0
1
1/4
1/2
2.25
1/16
4
5/32
9
1/32
Find probabilities given PMF
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Given a PMF, we can find the following
probabilities:
P(X=xi), P(x1<X<x2), P(X>xi) or
P(X<xi)
In those cases, we just find all X’s
whose values fall within the range and
add up the corresponding probabilities.
Find Probabilities given PMF
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In example 6: let’s find the following
probabilities:
1. P(X>0)
2. P(-1.8<X<2.5)
3. P(X<3)