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Probability Distributions
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Binomial:
The binomial distribution is a discrete distribution that is frequently
used to model the number of successes in a sample drawn with
replacement from a population.
- The experiment consists of n repeated trials.
- Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.
- The probability of success, denoted by p, is the same on every trial.
This implies that any selections are replaced.
- The trials are independent; that is, the outcome on one trial does not
affect the outcome on other trials.
The binomial probability is:
B(x,n,p) = _{n}C_{x} * P^{x} * (1 - p)^{n - x}
n = number of trials
x = number of successes
p = probability of success
q = probability of failure
_{n}C_{x} = the number of combination of n things taken x at a time
μ = np
σ^{2} = npq
Normal Approximation to Binomial:
If n is moderately large and p is not too extreme, then the binomial
distribution tends to be symmetric and is well approximated by a
normal distribution
mean = np
variance = npq
N(np,npq)
The normal distribution with mean np and variance npq can be used to
approximate a binomial distribution with parameters n and p when:
np and nq are both ≥ 10
Example 1.
Suppose a die is tossed 5 times. What is the probability of getting
exactly 2 fours?
P(2,5,0.167) = _{5}C_{2} * (0.167)^{2} * (0.833)^{3}
= 0.161
Example 2.
A barrel contains 6 good apples and 4 bad apples. If 3 apples are
pulled without replacement what is the probability distribution.
What is the mean and standard deviation?
GGG = (6/10)*(5/9)*(4/8) = 0.17
GGB = (6/10)*(5/9)*(4/8) = 0.17
GBB = (6/10)*(4/9)*(3/8) = 0.10
BBB = (4/10)*(3/9)*(2/8) = 0.03
BBG = (4/10)*(3/9)*(6/8) = 0.10
BGG = (4/10)*(6/9)*(5/8) = 0.17
BGB = (4/10)*(6/9)*(3/8) = 0.10
GBG = (6/10)*(4/9)*(5/8) = 0.17
Hypergeometric:
This is similar to the binomial distribution except that selections
are not replaced so the probability, p, changes from trial to trial.
The hypergeometric probability is:
h(x,N,n,k) = [_{k}C_{x}][_{N-k}C_{n-x}]/_{N}C_{n}
N = no. of items in population
k = number of items in population that are classified as successes
n = number of items in sample
x = number of items in sample that are classified as successes
_{k}C_{x} = The number of combinations of k things, taken x at a time
_{n}C_{x} = the number of combination of n things taken x at a time
μ = nk/N
σ^{2} = nk(N-k)(N-n)/[N^{2}(N-1)]
Example 1.
Select 5 cards w/o replacement from a deck of cards. What is probability
of getting 2 red cards.
N = 52
k = 26 there are 26 red cards in the decks
n = 5
x = 2
h(2,52,5,26) = [_{26}C_{2}][_{26}C_{3}]/_{52}C_{5}
= 0.325
Poisson:
The Poisson distribution is a discrete probability distribution that
expresses the probability of a given number of events occurring
in a fixed interval of time and/or space if these events occur with
a known average rate and independently of the time since the last
event. The Poisson distribution can also be used for the number
of events in other specified intervals such as distance, area or
volume.
- The experiment results in outcomes that can be classified as
successes or failures.
- The average number of successes (μ) that occurs in a
specified region is known.
- The probability that a success will occur is proportional to the
size of the region.
- The probability that a success will occur in an extremely small
region is virtually zero.
Note that the specified region could take many forms. For instance,
it could be a length, an area, a volume, a period of time, etc.
P(x,μ) = e^{-μ}(μ^{x}/x!)
μ = The mean number of successes that occur in a specified region
x = The actual number of successes that occur in a specified region
μ = μ
σ^{2} = μ
Example 1:
The average number of homes sold by the Acme Realty company
is 2 homes per day. What is the probability that exactly 3 homes
will be sold tomorrow?
μ = 2; since 2 homes are sold per day, on average.
x = 3; since we want to find the likelihood that 3 homes will be
sold tomorrow.
P(3,2) = (2.71828^{-2})(23)/3!
P(3,2) = (0.13534)(8)/6
P(3,2) = 0.180
Normal:
The binomial and Poisson distributions are examples of distributions
of a discrete random variable. The normal distribution is a
continuous distribution whose characteristics are often encountered
in practice.
N(μ,σ) = (1/√2πσ^{2})∫exp((x - μ)^{2}/2σ^{2})
Any normal distribution can be transformed to a standard normal
distribution using the z-transformation:
z = (x - μ)/σ
A Standard Normal distribution has mean 0 and variance 1. Areas
under this curve can be found using a standard normal table. In
these tables the total area under the curve is equal to 1.
Ex.
Suppose that SAT scores among U.S. college students are normally
distributed with a mean of 500 and a standard deviation of 100.
What is the probability that a randomly selected individual from
this population has an SAT score at or below 600?
z = (x - μ)/σ
= (600 - 500)/100
= 1
From the z score tables for -∞ to z we find that he probability
is 0.8413.