Download Discrete/Binomial Notes

Chapter 7
Discrete Distributions
Random Variable • A numerical variable
whose value depends
on the outcome of a
chance experiment
Two types:
• Discrete – count of some
random variable
• Continuous – measure of
some random variable
Discrete Probability Distribution
1) Gives the probabilities
associated with each possible x
value
2) Usually displayed in a table, but
can be displayed with a
histogram or formula
Discrete probability distributions
3)For every possible x value,
0 < P(x) < 1.
4) For all values of x,
S P(x) = 1.
Suppose you toss 3 coins & record
the number of heads.
The random variable X defined as ...
The number of heads tossed
Create a probability distribution.
X
P(X)
0
.125
1
.375
2
.375
3
.125
Create a probability histogram.
Let x be the number of courses for which
a randomly selected student at a certain
university is registered.
X
1 2 3
P(X).02 .03 .09
4 5 6 7
? .40 .16 .05
Why
does
this
not
start
at
zero?
.25
P(x = 4) =
P(x < 4) = .14
P(x < 4) = .39
P(x > 5) = .61
What is the probability that the student
is registered for at least five courses?
Formulas for mean & variance

  x
x 
x
2
xi p
i
i
 x  p
2
i
Found on formula card!
Let x be the number of courses for which
a randomly selected student at a certain
university is registered.
X
1
2
3
4
5
6
7
P(X).02 .03 .09 .25 .40 .16 .05
What is the mean and standard
deviations of this distribution?
  4.66
&
 = 1.2018
Here’s a game:
If a player rolls two dice
and gets aAsum
of 2is or
12,
fair game
one where
the cost to play EQUALS
he wins $20.theIf
he
gets
a
expected value!
0 $5. 5The cost
20
7,X he wins
to
P(X)
7/9
1/6
1/18
roll the dice one time is
$3. Is this game fair?
NO, since  = $1.944 which is less
than it cost to play ($3).
Linear transformation of a
random variable
The
mean
is
changed
If x is a random variable, then the
by
addition
&
random variable y is defined by
multiplication!
y  a  bx The standard
deviation is
and
ONLY changed
 y  a  bx
by
multiplication!
  b
y
x
Let x be the number of gallons
required to fill a propane tank.
Suppose that the mean and
standard deviation is 318 gal. and 42
gal., respectively. The company is
considering the pricing model of a
service charge of $50 plus $1.80
per gallon. Let y be the random
variable of the amount billed. What
is the mean and standard deviation
for the amount billed?
 = $622.40 &  = $75.60
JustLinear
add or subtract
combinations
the means!
 a b   a  b
 a b   a   b
2
a
 a b    
2
b
If independent, always add
the variances!
A nationwide standardized exam consists of a
multiple choice section and a free response
section. For each section, the mean and
standard deviation are reported to be
mean
SD
MC
38
6
FR
30
7
If the test score is computed by adding the
multiple choice and free response, then what is
the mean and standard deviation of the test?
  68
&
 = 9.2195
Special Discrete
Distributions
Binomial Distribution B(n,p)
1. Each trial results in one of two
mutually exclusive outcomes.
(success/failure)
2. There are a fixed number of trials
3. Outcomes of different trials are
independent
4. The probability that a trial results in
success is the same for all trials
The binomial random variable x is
defined as the number of successes out
of the fixed number
Are these binomial distributions?
1) Toss a coin 10 times and count the
number of heads
Yes
2) Deal 10 cards from a shuffled deck
and count the number of red cards
No, probability does not remain constant
3) Two parents with genes for O and A
blood types and count the number of
children with blood type O
No, no fixed number
Binomial Formula:
n  k
n k
P (x  k )    p 1  p 
k
 
Where:
n 
 n C k
k 
Out of 3 coins that are tossed,
what is the probability of
getting exactly 2 heads?
B (3,.5)
3 2
1


P (x  2)   0.5 0.5  .375
2
The number of inaccurate gauges in a
group of four is a binomial random
variable. If the probability of a defect
gauge is 0.1, what is the probability
that only 1 is defective?
B ( 4,.1)
 4
P (x  1)   0.11 0.93  .2916
1 
More than 1 is defective?
P (x  1)  1  (P (0)  P (1))  .0523
Calculator
• Binomialpdf(n,p,x) – this calculates
the probability of a single binomial
P(x = k)
• Binomialcdf(n,p,x) – this calculates
the cumulative probabilities from
P(0) to P(k) OR
P(X < k)
A genetic trait of one family
manifests itself in 25% of the
offspring. If eight offspring are
randomly selected, find the
probability that the trait will
appear in exactly three of them.
B (8,.25)
P (X  3)  binompdf (8,.25,3)  .2076
At least 5?
P (X  5)  1  binomcdf (8,.25,4)  .0273
In a certain county, 30% of the voters
are Republicans. If ten voters are
selected at random, find the probability
that no more than six of them will be
Republicans.
B(10,.3)
P(x < 6) = binomcdf(10,.3,6) = .9894
What is the probability that at least 7
are not Republicans?
P(x > 7) = 1 - binomcdf(10,.7,6) = .6496
Binomial formulas for mean
and standard deviation
 x  np
 x  np 1  p 
In a certain county, 30% of the
voters are Republicans. How
many Republicans would you expect
in ten randomly selected voters?
What is the standard deviation for
this distribution?
B (10,.3)
x  10(.3)  3 Republicans
 x  10(.3)(.7)  1.45 Republicans
In a certain county, 30% of the
voters are Republicans. What is the
probability that the number of
Republicans out of 10 is within 1
standard deviation of the mean?
B(10,.3)
P(1.55 < x < 4.45) = binomcdf(10,.3,4) –
binomcdf(10,.3,1) = .7004
Geometric Distributions:
1. There are two mutually
So what are the
exclusive
outcomes
How
far
this go?
To will
infinity
possible values of X
2. Each trial is independent of the
others
3. The probability of success
remains constant for each trial.
X random
.
1 2 variable
3 4 x.is. the
The
number of trials UNTIL the
FIRST success occurs.
Differences between binomial
& geometric distributions
• The difference between
binomial and geometric
properties is that there is
NOT a fixed number of
trials in geometric
distributions!
Other differences:
•Binomial random variables
start with 0 while geometric
random variables start with 1
•Binomial distributions are
finite, while geometric
distributions are infinite
Geometric Formulas:
P (x )  p 1  p 
x 1
1
x 
p
1p
x 
2
p
Not on formula
sheet – they will
be given on quiz or
test
Count the number of boys in a
family of four children.
Binomial:
X
0
1
2
3
What are the
values for these
random
4
variables?
Count children until first son
is born
Geometric:
X
1
2
3
4
. . .
Calculator
• geometpdf(p,x) – finds the
geometric probability for P(X = k)
No “n” because
there is no
• Geometcdf(p,x) – finds the
fixed number!
cumulative probability for P(X < k)
What is the probability that the
first son is the fourth child born?
P (X  4)  geometricpdf (.5,4)  .0625
What is the probability that the first
son born is at most the fourth child?
P (X  4)  geometriccdf (.5,4)  .9375
What is the probability that the first
son born is at least the third child?
P (X  3)  1  geometriccdf (.5,2)  .25
A real estate agent shows a house to
prospective buyers. The probability that
the house will be sold to the person is
35%. What is the probability that the
agent will sell the house to the third
person she shows it to?
P (x  3)  geometricpdf (.35,3)  .1479
How many prospective buyers does she
expect to show the house to before
someone buys the house? SD?
1
1  .35
x 
 2.86 buyers  x 
 2.304 buyers
2
.35
.35