Probability distribution 10
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The document outlines two main types of probability distributions: discrete (e.g., binomial, Poisson) and continuous (e.g., normal distribution). It emphasizes the significance of the normal distribution in real-world situations and statistical inference, highlighting its characteristics such as symmetry and specific area coverage under the curve. Additionally, the document describes the binomial and Poisson distributions, detailing their assumptions and applications in various contexts.
Probability distribution 10
- 2. Probability Distribution There are two types of Probability Distribution; 1) Discrete Probability Distribution- the set of all possible values is at most a finite or a countable infinite number of possible values Poisson Distribution Binomial Distribution 1) Continuous Probability Distribution- takes on values at every point over a given interval Normal (Gaussian) Distribution
- 3. Normal (Gaussian) Distribution • The normal distribution is a descriptive model that describes real world situations. • 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). • This is the most important probability distribution in statistics and important tool in analysis of epidemiological data and management science. Characteristics of Normal Distribution • It links frequency distribution to probability distribution • Has a Bell Shape Curve and is Symmetric • It is Symmetric around the mean: Two halves of the curve are the same (mirror images) • Hence Mean = Median • The total area under the curve is 1 (or 100%) • Normal Distribution has the same shape as Standard Normal Distribution. • In a Standard Normal Distribution: The mean (μ ) = 0 and Standard deviation (σ) =1
- 4. Normal (Gaussian) Distribution(2) Z Score (Standard Score) • Z = X - μ • Z indicates how many standard deviations away from the mean the point x lies. • Z score is calculated to 2 decimal places. Tables Areas under the standard normal curve
- 5. 13.6% 2.2% 0.15% -3 -2 -1 μ 1 2 3 Diagram of Normal Distribution Curve (z distribution) 33.35%
- 6. Normal (Gaussian) Distribution(4) Distinguishing Features • The mean ± 1 standard deviation covers 66.7% of the area under the curve • The mean ± 2 standard deviation covers 95% of the area under the curve • The mean ± 3 standard deviation covers 99.7% of the area under the curve Application/Uses of Normal Distribution • It’s application goes beyond describing distributions • It is used by researchers and modelers. • The major use of normal distribution is the role it plays in statistical inference. • The z score along with the t –score, chi-square and F-statistics is important in hypothesis testing. • It helps managers/management make decisions.
- 7. Binomial Distribution A widely known discrete distribution constructed by determining the probabilities of X successes in n trials. Assumptions of the Binomial Distribution • The experiment involves n identical trials • Each trial has only two possible outcomes: success and failure • Each trial is independent of the previous trials • The terms p and q remain constant throughout the experiment – p is the probability of a success on any one trial – q = (1-p) is the probability of a failure on any one trial • In the n trials X is the number of successes possible where X is a whole number between 0 and n. • Applications – Sampling with replacement – Sampling without replacement causes p to change but if the sample size n < 5% N, the independence assumption is not a great concern.
- 8. Binomial Distribution Formula • Probability function • Mean value • Variance and standard deviation P X n X n X X n X n X p q( ) ! ! ! for 0 n p 2 2 n p q n p q
- 9. Poisson Distribution French mathematician Siméon Denis Poisson proposed Poisson DistributionThe Poisson distribution is popular for modelling the number of times an event occurs in an interval of time or space. It is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. The Poisson distribution may be useful to model events such as • The number of meteorites greater than 1 meter diameter that strike Earth in a year • The number of patients arriving in an emergency room between 10 and 11 pm • The number of photons hitting a detector in a particular time interval • The number of mistakes committed per pages
- 10. Poisson Distribution Assumptions of the Poisson Distribution • Describes discrete occurrences over a continuum or interval • A discrete distribution • Describes rare events • Each occurrence is independent any other occurrences. • The number of occurrences in each interval can vary from zero to infinity. • The expected number of occurrences must hold constant throughout the experiment.
- 11. © 2002 Thomson / South-Western Slide 5-11 Poisson Distribution Formula • Probability function P X X X where long run average e X e( ) ! , , , ,... : . ... for (the base of natural logarithms) 0 1 2 3 2 718282 Mean value Standard deviation Variance