Download Probability and Probability Distributions - Statistical Analysis | STAT 401 and more Study notes Statistics in PDF only on Docsity!
CHAPTER 4: PROBABILITY AND
PROBABILITY DISTRIBUTIONS
E F , : any two events (#E is the complement of E)
Additive law
T Ea or F b œ T Ea b & T Fa b # T Ea andFb
Multiplicative law
T Ea andF b œ T ElF T Fa b a b œ T FlE T Ea b a b
Independence
E and Fare independent if
T ElFa b œ T Ea b
Bayes' formula
T ElF œ
T FlE T E T F
a b
a b a b a b
This is Bayes' formula. Often the denominator probability must be obtained as a sum of joint probabilities, for instance, the event Fcan be written as the union of a F and Eb and a F and E#b:
So
T F œ T F E & T F E
a b a and b a and b
œ T FlE T E a b a b & T FlE T Ea #^ b a# b
and
T ElF œ
T FlE T E T FlE T E & T FlE T E#^ #
a b
a b a b a b a b a b a b
Probability of two socks the same color:
T [a (^) " and [ (^) # b & T Fa (^) " and F (^) #bœ & œ
Probability that the second sock is blue:
T Fa (^) # bœT F l[a (^) # " bT [ a (^) " b & T F lFa (^) # " bT F a (^) "b
œ † & † œ
Probability that the first sock is white, given the second sock is blue:
T [ lF œ
T F l[ T [ a b
a b a b " #
" "
T Fa (^) #b
(Bayes' rule)
œ
T F l[ T [ T F l[ T [ & T F lF T F
a b a b a b a b a b a b
" "
" " # " "
2 1 3 2 2 1 1 1 3 2 3 2
œ
Discrete probability distributions
Random variable: a numerical outcome of a random experiment (usually denoted with an upper case letter, e.g. ] ) ex. # of blue socks SAT of a randomly drawn student
democrats in a random sample of voters
1 day's growth (dry weight) of a plant
Probability distribution: collection of all possible outcomes of a random variable, and their associated probabilities (a particular outcome usually denoted with a lower case letter, e.g., C)
Discrete probability distribution: random variable has a finite or countably infinite number of states
The expected value or mean of the binomial random variable ]:
E a ] b œ .œ 0 † T a b 0 & 1 † T a b 1 & â & 8 † T 8a b
ALA^ œ^81
Variance of the random variable ]:
V a ] b œ 5 #^ œ a 0 # .b #^ T a b 0 & a 1 # .b #T a b 1 & â
& a8 # .b #^ T 8a b (^) ALAœ 8 1 a 1 # 1 b
Common notation: ] μ binomial a 8 , 1 b
ì ] has a binomial distribution with parameters 8 and 1 î
SAS:
The function RANBIN(seed, n, p) returns a binomial random variable with # trials n and success probability p (set seed = 0 and SAS will use the computer clock time as seed)
The function PROBBNML(p, n, x) computes the probability that an observation from a binomial(n, p) distribution will be less than or equal to x.
exercise (concept of mean and variance of a discrete distribution): suppose ] has a ìrectangular distributionî given by
T ] œ Ca b œ T Ca b œ
C œ 1, 2, 3, 4, 5, 6
(distribution of the result of rolling a die). (a) Draw a picture of the probability distribution. (b) Calculate the expected value of ]. (c) Calculate the variance of ].
2. Poisson distribution
T ] œ C œ T C œ
Cx
a b a b
#. .C
C œ 0, 1, 2, 3, á.
Here / œ 2.71828... and. is a parameter (. 5 0). This is a distribution with positive probability on all the nonnegative integers:
T a b 0 & T a b 1 & T a b 2 & â œ "T Ca b œ1. Cœ!
_
Suppose 8 is large and 1 is small. Then
T ] œ C ¸ œ 8 Cx
a b
e where
#. .C
(Poisson approximation to the binomial)
SAS:
RANDPOI(seed, m) generates a Poisson random variable with mean m
POISSON(m, x) calculates T ] Ÿa x , whereb ] μ Poissona b.
3. Multinomial distribution (a multivariate distribution)
5 types
♦ ♦ æ ♥ ♥ sample 8 ♦ ♥ ♦ æ with replacement æ ♦ ♥ ♥ ♥
(^1) " œ proportion of type 1 (æ ) in the urn (^1) # œ proportion of type 2 ( ♥) in the urn ã 15 œ propostion of type 5 ( ♦) in the urn
The 14 's are constants (parameters); (^1) " & (^1) # & â & 15 œ 1.
] (^) " , ] (^) # , á , ] 5 : random variables
] (^) " œ number of type 1 (æ ) in the sample
] (^) # œnumber of type 2 (♥ ) in the sample ã
] 5 œ number of type 5 ( ♦) in the sample
] (^) " & ] (^) # & â & ] 5 œ 8.
The ] 4 's are dependent : the value of one affects the others.
