MA 320-001: Introductory Probability
David Murrugarra
Department of Mathematics,University of Kentucky
http://www.math.uky.edu/~dmu228/ma320/
Spring 2017
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 1 / 18
The Uniform Distribution
Let the random variable X denote the outcome when point is selectedat random from an interval [a,b],−∞ < a < b <∞. If the experimentis performed in a fair manner, it is reasonable to assume that theprobability that the point is selected from the interval [a, x ],a ≤ x < b,is (x − a)/(b − a). That is, the probability is proportional to the lengthof the interval, so the distribution function of X is
F (x) =
0, x < a,x−ab−a , a ≤ x < b,1, b ≤ x .
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 2 / 18
The Uniform Distribution
Because X is a continuous-type random variable, F ′(x) is equal to thep.d.f. of X whenever F ′(x) exists; thus, when a < x < b, we havef (x) = F ′(x) = 1/(b − a).
The random variable X has a uniform distribution if its p.d.f. is equalto a constant on its support. In particular, if the support is the interval[a,b], then
f (x) =1
b − a, a ≤ x ≤ b.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 3 / 18
Uniform Distribution PDF
Figure: Uniform Distribution PDF
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 4 / 18
Uniform Distribution CDF
Figure: Uniform Distribution c.d.f.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 5 / 18
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 6 / 18
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 6 / 18
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.
A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.
∑x∈S
f (x) = 1
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 6 / 18
Binomial Distribution
Figure: Binomial density function.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 7 / 18
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
µ = E(X ) = np.
σ2 = npq
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 8 / 18
Binomial Distribution
f (x) =(
nx
)px(1− p)n−x , x = 0,1,2, . . . ,n.
µ = E(X ) = np.
σ2 = npq
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 8 / 18
Cumulative Distribution Function
The cumulative distribution function or, more simply, thedistribution function of the random variable X is
F (x) = P(X ≤ x), −∞ < x <∞,
For the binomial distribution the distribution function is defined by
F (x) = P(X ≤ x) =bxc∑y=0
(ny
)py (1− p)n−y
where bxc is the floor or greatest integer less than or equal to x .
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 9 / 18
Binomial Distribution
Figure: Binomial distribution cdf
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 10 / 18
The Poisson Distribution
DefinitionLet the number of changes that occur in a given continuous interval becounted. Then we have an approximate poisson process withparameter λ > 0 if the following conditions are satisfied:
1 The number of changes occurring in nonoverlapping intervals areindependent.
2 The probability of exactly one change occurring in a sufficientlyshort interval of length h is approximately λh.
3 The probability of two or more changes occurring in a sufficientlyshort interval is essentially zero.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 11 / 18
The Poisson Distribution
The random variable X has a Poisson distribution if its densityfunction is of the form
f (x) =λxe−λ
x!, x = 0,1,2, ...,
where λ > 0.
In this case, µ = σ2 = λ.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 12 / 18
The Poisson Distribution
The Poisson distribution has been used to model:
1 The number of chocolate chips in a cookie.2 The number of calls coming into a call centre.3 The number of deaths from horse kicks in the Prussian army
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 13 / 18
The Poisson Distribution
Figure: Poisson distribution pmf
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 14 / 18
The Poisson Distribution
Figure: Poisson distribution cdf
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 15 / 18
The Poisson Distribution
If events in a poisson process occur at a mean rate of λ per unit, theexpected number of occurrences in an interval of length t is λt .
For example, if phone calls arrive at a switchboard following a Poissonprocess at a mean rate of three per minute, then the expected numberof phone calls in a 5-minute period is (3)(5) = 15.
Moreover, the number of occurrences say, x , in the interval of length thas the Poisson density function,
f (x) =(λt)xe−λt
x!, x = 0,1,2, ...
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 16 / 18
The Poisson Distribution
ExampleTelephone calls enter a college switchboard on the average of twoevery 3 minutes. If one assumes an approximate Poisson process,what is the probability of five or more calls arriving in a 9 minutesperiod?
Let X denote the number of calls in a 9 minute period.
David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 17 / 18
The Poisson Distribution
ExampleTelephone calls enter a college switchboard on the average of twoevery 3 minutes. If one assumes an approximate Poisson process,what is the probability of five or more calls arriving in a 9 minutesperiod?
Let X denote the number of calls in a 9 minute period.
We see that E(X ) = 6; that is, on the average, six calls will arriveduring a 9 minute period. Thus,
P(X ≥ 5) = 1− P(X ≤ 4) = 1−4∑
x=0
6xe−6
x!
= 1− 0.285 = 0.715David Murrugarra (University of Kentucky) MA 320: Section 5.1 Spring 2017 18 / 18