• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Law of large numbers wikipedia, lookup

Probability interpretations wikipedia, lookup

Ars Conjectandi wikipedia, lookup

Inductive probability wikipedia, lookup

Birthday problem wikipedia, lookup

Randomness wikipedia, lookup

Infinite monkey theorem wikipedia, lookup

Conditioning (probability) wikipedia, lookup

Indeterminism wikipedia, lookup

Random variable wikipedia, lookup

Transcript
```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 random variable X defined as ...
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
&
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
combinations
the means!
 a b   a  b
 a b   a   b
2
a
 a b    
2
b
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
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
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
•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
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