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Probability DistributionsDanstan Bagenda PhD

Makerere University - School of Public Health

24th September 2007

Variable

• The characteristic of interest in a study if called a VARIABLE eg; weight of students in class

• Term VARIABLE makes sense becoz:

• value varies from subject to subject

• variation results from inherent biological variation among individuals

• errors made in measuring & recording subjects value on a characteristic

Random Variable

• variable in a study in which subjects are RANDOMLY selected

• outcome of a RANDOM EXPERIMENT

• EG; draw 2 student’s at RANDOM from class. Thats a random expt.

• student’s weights, heights, family incomes etc... of RANDOMLY selected students are ALL RANDOM VARIABLES

Random Variable

• TOSS TWO COINS: (The Random Expt)

• Record the NO. OF HEADS (x) : 0, 1, or 2 (Random Variable)

Outcome TT HT or TH HH

x 0 1 2

Probability Distribution• PROBABILITIES of the outcomes

• Probability that the RV X has the value x

• Pr(X=x)= p(x)

x 0 1 2

P(X=x) 1/4 1/2 1/4

Probability Distributions

• In some applications, a formula or rule will adequately describe the distribution

• In other situations, a theoretical distribution provides a good fit to the variable of interest

Probability Distributions

• Several THEORETICAL probability distributions are important

• we examine 3 of importance in medicine & public health:

• Binomial & Poisson - Discrete (associated rv takes on ONLY integer values 0,1,2,...,n)

• NORMAL (gaussian) - Continuous - rv’s measured on continuous scale

Binomial Distribution

• Event has only TWO possible OUTCOMEs

• eg: Head or Tails

• Success or Failure

• Characterized by 2 parameters:

• n = no. of independent trials

• p= probability of success of each trial

Binomial Distribution• Basic principles developed by Swiss

mathematician Jacob Bernoulli (1713)

• A repeatable Expt called a BERNOULLI TRIAL provided:

• 1) the result of each trial may either be a success or a failure

• 2) the probability p of success is the SAME in EVERY TRIAL

• 3) The trials are INDEPENDENT: the outcome of 1 trial has no influence on later outcomes

Binomial Distribution• TOSS TWO COINS: (The Random Expt)

• Record the NO. OF HEADS (x) : 0, 1, or 2 (Random Variable)

• here: n= 2, no. of successes (x) & p=0.5

Outcome TT HT or TH HH

x 0 1 2


x 0 1 2

P(X=x) 1/4 1/2 1/4



nCi = n!i!(n−i)!


Binomial Distribution Binomial Distribution

Binomial DistributionBinomial Distribution

Binomial Distribution Binomial Distribution

Poisson Distribution• Named after French mathematician who

derived it Simeon D. Poisson

• Like Binomial, it is DISCRETE

• Used to determine probability of RARE events

• Similar to Binomial except that n (no of trials is very large) & p (probability of success is very small)

• no. of beds hospital needs in its ICU

• no. of cells in a given volume of fluid

• no. of bacterial colonies growing in a given medium

Poisson Distribution• RV - no. of times an event occurs in a given

time or space interval

• Probability of exactly x occurrences is given by:

• ! is value of BOTH the mean & variance of the Poisson distribution, &

• e is the base if the natural log (=2.718)

• NOTE: while binomial distribtn has 2 parameters “n” & “p”, poisson ONLY needs 1 “!”.

P (X) = !xe−λ

X!

Poisson Distribution• Eg: No. of hospitalizations for patients

following Coronary Artery Surgery

• Model appropriate because the chance that a patient goes into hospital in any time interval is small & can be assumed indpt from patient to patient.

• 390 patients hospitalized a total of 660 times over a 5-year period. Thus, mean no. of hospitalizations (!) is 660/390=1.69

• The prob. that a patient has zero hospitalizations is:

• P(X=0) = = 0.184P (X = 0) = 1.690e!1.69

0!

Normal (Gaussian) Distrib.

• Discovered by French mathematician Abraham Demoivre (1733)

• Scientific principles by 2 astronomers:

• Pierre-Simon Laplace (France) & Carl Friedrich Gauss (Germany) - theory of motion of heavenly bodies

• Continuous RV: Can take on any value

• Symmetric about mean

Normal Distribution Normal Distribution


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