Transcript
PROBABILITY DISTRIBUTION
Overview
• Types • Importance• Properties• Area under normal distribution• Standard normal curve• Z transformation• Calculating the areas• Application
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Introduction
• Probability : likelihood of occurrence of an event
• Probability distribution : is a mathematical representation of the probabilities associated with the values of random variable
• Random variable : variable whose value is determined by outcome of a random experiment
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I. Discrete probability distributions : Binomial distribution Poisson distribution
II. Continuous probability distributions : Normal distribution
Types
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Binomial distribution
• Bernoulli distribution
• Bernoulli process/trial : is one where an experiment can result only in one or two mutually exclusive outcomes.
• Binomial distribution : distribution of probabilities where there are only two possible outcomes for each trail of an experiment.
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Binomial distribution
Assumptions : Each trial has only two possible outcomes. Probability of success or failure remains constant from trail to trail.
In n trials, success denoted by r, and failure by n – r; probability ( r successes out of n trials ) :
p(r) = [n! / r!(n – r)!] . pr qn-r
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Poisson distribution
• Limiting the distribution of binomial distribution where number of trials, n, is very large and the probability, p, of success for every trial is very small.
• ‘np’ is the fixed number which is called poisson distribution.
• This distribution studies the probabilities of rare events which are common in science.
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Poisson distribution
• Ex : no of defective articles produced by a high quality machine
• Probability of r successes :
p(r) = e-m mr / r!
where r = 0, 1, 2,….. N successes e = 2.7183 (constant)
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Normal distribution• First discovered by De Moivre
• Also by Laplace and Guass
• The Normal distribution is also known as the Gaussian Distribution and the curve is also known as the Gaussian Curve.
• Named after German Mathematician Astronomer Carl Frederich Gauss.
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Normal distribution
• It is defined as a continuous frequency distribution of infinite range (can take any values not just integers as in the case of binomial and Poisson distribution).
• Curve with important statistical properties, having a smooth curve with symmetrical distributed items on both sides of the peak, is called normal distribution curve.
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Importance
• In biological analyses
• Sample size too large-normal distribution serves a good approximation of discrete distribution.
• Making references regarding the value of population mean from sample mean.
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Properties
• Bell shaped and symmetrical• Single peak-unimodal• Mean lies at the centre• Mean, median and mode are all equal• The total area under the curve the same as any other
probability distribution is 1 or (100%)12
• Height declines at either side of the peak• Height max at the mean• Area on both sides is equal to each other• Curve is asymptotic to the base on either side• Probabilities for the normal random variable are
given by areas under the curve.
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Changing μ shifts the distribution left or right.
Changing σ increases or decreases the spread.
µ
Normal curves are defined by two parameters :• Mean : measure of location• Standard deviation : measure of spread
Variations with mean and standard deviation
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Same mean with different standard deviation
Same standard deviation withdifferent mean
Different mean values with different standard
deviation
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Area under normal distribution
• ± 1s covers 68.27% area; 34.14% area lie on either side of the mean
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• ± 2s covers 95.45% area; 47.73% area lie on either side of the mean
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• ± 3s covers 99.73% area; 49.87% area lie on either side of the mean
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Area under normal distribution
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No matter what and are, the area between - and + is about 68%;
the area between -2 and +2 is about 95%;
the area between -3 and +3 is about 99.7%.
Almost all values fall within 3 standard deviations.
Standard normal curve
Curve with zero mean and unit standard deviation
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Z transformation• Normal distribution : z = x – / s (sample) z = x – μ / σ (population)
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Table of cumulative areas under standard normal cuvre
z = x-μ/σ Area from -∞ to Z-2.576 0.005
-2.326 0.01
-1.96 0.025
-1.645 0.05
-1.58 0.057
-1.28 0.10
-1 0.16
-0.5 0.31
0 0.5023
Table of cumulative areas under standard normal cuvre
z = x-μ/σ Area from -∞ to Z0.50 0.69
1.00 0.84
1.28 0.90
1.58 0.943
1.645 0.95
1.96 0.975
2.326 0.99
2.567 0.995
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Calculating the areas• Probability of choosing a tablet at random that
weighs between 190 and 210 mg ?
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Calculating the areas• To calculate the area between 190 and 210 :
a. Area between -∞ and 210 : z = x-μ/σ = 210-200/10 = 1
From the table area between -∞ and 210 is 0.84.
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b. Area between -∞ and 190 is 0.16 z = 190-200/10 = -1
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c. Area between 190 and 210 is 0.68 = (area between -∞ and 210) – (area between -∞ and
190 is 0.16)
• Probability of choosing a tablet at random between 190 and 210 mg is 0.68 or 68%
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Practical Application
• According to USP, for tablets of weight 100mg : not more than 2 tablets – may deviate by more
than 10% (in a batch of 20 tablets) no tablet must differ by more than 20%
• For passing this test, 98% of the tablets must weight within 10% of the mean
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Practical Application
• So, 1000 tablets from a batch of 30,00,000 tablets are weighed and mean and standard deviation are calculated with following formulae's :
• Mean = 101.2, standard deviation = 3.92
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Practical Application
• Within 10% of mean (101.2) = 91.1 and 111.3
• We should find out what probability of tablets is between 91.1 and 111.3? (If 98% or more, USP requirements are met)
• For this, a normal distribution is constructed and areas are found out.
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a) Area between -∞ and 111.3 z = x-μ/σ = 111.3 – 101.2 / 3.92 = 2.58From table, area is 0.995
b) Area between -∞ and 91.1 z = 91.1 – 101.2 / 3.92 = -2.57 From table, area is 0.005
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c) Area between 91.1 and 111.3 = 0.995 – 0.005 = 0.99
Thus probability of tablets is between 91.1 and
111.3 is 0.99 or 99% Hence passes the test
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References
• Khan and Khanum, Fundamentals of biostatistics, 3rd edition, Ukaaz Publicaions, pg no. 181
• Leon Lachman and Herbert A. Lieberman, The theory and practice of Industrial Pharmacy, 2009 edition, pg no. 246
• Sanford Bolton, Charles Bon, Pharmaceutical Statistics: Practical and clinical applications, 4th edition, volume 135, pg no. 54
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