Lecture 12
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Probability Distribution for Random Variables
Next few classes - probability distribution applied to quantitative variables
The outcomes for the random trials of interest for such problems are called Random Variables
Random Variables maybe Discrete or Continuous
Notations and procedures are somewhat different depending on whether we have discrete or continuous variables
Probability Distribution for Discrete Random Variable
Example: Select tow students at random. Variable of interest is X, the no. from out of state
x: 0 1 2 P(x): 0.74 0.24 0.02 Note: Upper case, X, refers to the variable - lower case, x, refers to a specific outcome for the variable
What is P(X=1) or P(1)?
P(X<1)?
P(X< or = 1)? P(X<2)?
Bivariate Probability Distribution
Example: X - number from out of state, Y - number of males
Number of Males (Y) Number out of state (X) 1 2 3 Total 0 0.01 0.08 0.65 0.74 1 0.00 0.02 0.22 0.24 2 0.00 0.00 0.02 0.02 Total 0.01 0.10 0.89 1.00
Are the two variables independent?
Indep. if Joint Prob of x and y = P(x)P(y)
Expected Value and Variance
Expected value - average outcome of many trials
x: 0 1 2 P(x): 0.74 0.24 0.02 What is expected value? 0? 1? 2? or near 1?
0.28
E{X} = sum of x times P(x)
Variance of X
Will the variance be small for the above example?
s2{X} = sum of (x - E{X}) squared times P(x)
Lecture 13
Binomial Probability Distribution
A binomial probability distribution gives the probability associated with each outcome in a binomial experiment
The binomial experiment can be used to model many types of real life problems
How do we know when an experiment is binomial?
A binomial experiment consists of a series of Bernoulli trials
A Binomial Experiment
What is a Bernoulli trial?
Two possible outcomes - yes/no, pass/fail
one outcome assigned value X=1 the other X=0
The probability of a given outcome is always the same for each trial
P(X=1) = p P(X=0) = 1 - p
Each trial is statistically independent
The random variable of interest for the binomial experiment is the number of successes (or failures) during ‘n’ trials
Binomial Example
Example: Three randomly selected specimens are tested to see if they meet minimum strength. The probability that a given specimen will meet the requirement is 0.90. What is the probability that only two specimens will pass the test.
Is this a binomial experiment?
Two outcomes
Probability is same for each trial
Statistical independence
Random variable is number of successes.
Probability of success?
Sample space?
Binomial Probability Function
Where P(x) = P(X=x)
x = 0,1,…,n
n - number of trials
p - probability of X=1
binomial coefficient
e.g.
= 4.3.2.1/3.2.1.1 = 4
Concrete Test
Probability of two specimens passing test
p? n?
We need probability of P(X=2)
= 3*0.81*0.1 = 3*0.081 = 0.243
What do the 3 and the 0.081 represent?
110, 101, 011
probability of 110?
Expected Value and Variance
E{X} = np
s2{x} = np(1-p)
Lecture 14
Poisson Distribution
Poisson distribution apply to many random phenomena occurring in space or time
For example
Number of arrivals in a second
number of microbes in a cubic feet of water
Conditions for Poisson
The experiment consists of counting the number of times an event occur during a unit of time or a given area or volume
The probability that the event occur in a given unit is the same for all units
The number of events in one unit is independent of the number in other units
Poisson Probability Function
P(x) = lx exp(-l)/x!
Note: exp(-l) = e-l
Mean and Variance of the Poisson Distribution
Mean, E{X} = l
Variance, s 2{X} = l
l is the mean (average number in a time period)
Poisson is a good approximation to binomial if ‘p’ is small, n is large and pn < 5
Poisson Example
The number of cars arriving at a garage can be modeled by a Poisson distribution with mean of 0.8 cars/s
Find a) P(X=0)
a) P(X=0) = 0.80 exp(-0.8)/0! = 1*0.449/1
= 0.449
Complete the probability distribution for this problem. Use Table III to varify your results.
Use the probability distribution to find b) P(X=2), c) probability that more than 2 cars arrive per second
b) P(X=2)?
c) prob(more than 2)?
P(X>2) = P(3) + P(4) + P(5) + ….
or 1 - P(X<=2) = 1 - P(0) - P(1) - P(2)
Be careful with < versus <=
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Date of last update - 05 Oct 1998