Probability distribution binomial Poisson and normal.pptx

Probability distribution binomial Poisson and normal.pptx

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  Probability Distributions (Binomial, Poisson & Normal) DrShowkat Ahmad Wani
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  What is aProbability Distribution? • A probability distribution tells us how the probabilities of different outcomes are spread out for a random variable. • In simple words: If you have an experiment that can give different results, a probability distribution tells you how likely each result is. Examples in biology/parasitology: • Number of infected individuals in a sample • No. of Parasites in a host • No. of eggs per gram of Faeces. • Success/Failure of a diagnostic test.
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  • Different typesof data follow different patterns. If we know the pattern, we can choose the correct: • Statistical test • Method of analysis • Interpretation of results • Without this, research can go wrong because wrong distribution → wrong test wrong conclusion → . • Example: If parasite egg counts follow Poisson, but you analyze them as Normal, your conclusions become false.
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  Binomial Distributio n • When toUse It? • Use it when: • Only two outcomes exist Yes/No, Positive/Negative → • Number of trials (n) is fixed • Probability of success (p) is constant • Trials are independent • Simple Parasitology Example • A child is either: • Infected (Success) • Not infected (Failure) • If you test 100 children for Ascaris, the pattern of infected/not infected follows a binomial distribution. • Why is it important for researchers? • It helps answer: • What is the probability that 30 out of 100 children are infected? • Is the infection rate higher than expected? • Is a diagnostic test performing correctly? • You cannot correctly calculate prevalence or compare infection rates without using binomial principles.
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  Binomial Distributio n (n =10, p = 0.5) • Explanation of Parameters • n = number of trials Here, n = 10 means we perform the experiment → 10 times (e.g., testing 10 children). • p = probability of success p = 0.5 means the chance of success in each trial is → 50% (e.g., chance a child is infected = 0.5). • Meaning of the Diagram • The bars show the probability of getting 0, 1, 2, …, 10 successes out of 10 trials. • The highest bar is around 5 successes, because when p = 0.5, getting half successes is most likely. • The shape is symmetric when p = 0.5. • Simple Biological Example • If infection chance is 50%, the diagram shows the probability of having: • 0 infected children • 1 infected child • 2 infected children … • up to 10 infected children • out of 10 tested.
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  Poisson Distributio n • When toUse It? • Use Poisson when: • You are counting rare, random events • They occur independently • They occur at a constant average rate (λ) • Simple Parasitology Example • Hookworm eggs per gram of stool: • Some children have 0 • Some have 1 or 2 • Rarely someone may have 5 or more • This fits a Poisson distribution. • Why is it important for researchers? • It helps you decide: • Is the number of eggs increasing normally or due to an outbreak? • Are infections randomly spread or clustered? • Is the transmission stable? • You cannot correctly analyze egg counts, case counts, or rare parasitic events without Poisson distribution.
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  Poisson Distributio n (λ =3) • Explanation of λ (Lambda) • λ (lambda) = average rate of occurrence Here, λ = 3 means → the average number of events is 3 (e.g., on average 3 hookworm eggs per gram of stool). • Meaning of the Diagram • The bars show the probability of observing 0, 1, 2, 3, 4, … eggs per gram. • The highest probability is at k = 3, because that is the average. • The distribution is skewed, not symmetric. • Simple Biological Example • If the average egg count is 3 eggs/g, the diagram shows the chance of finding: • 0 eggs • 1 egg • 2 eggs • 3 eggs • up to 14 eggs • in a sample. Because egg counts are random and rare, Poisson fits this well.
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  Normal Distributio n • When toUse It? • Use it for continuous biological measurements that show a bell-shaped curve. • Simple Biological Example • Hemoglobin (Hb) levels of 300 children: • Most children have Hb near the mean (e.g., 11 g/dL) • Few have very high or very low values • This typically forms a normal distribution. • Why is it important for researchers? • Normal distribution is the foundation of: • Mean, SD • t-test • z-test • Confidence intervals • Regression analysis • Without understanding normal distribution, you cannot correctly compare biological measurements.
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  Normal Distributio n (μ =0, σ = 1) • Explanation of Parameters • μ (mu) = mean Here, μ = 0 means the average value is 0. → • σ (sigma) = standard deviation σ = 1 means values typically lie within → ±1 of the mean for 68% of the cases. • Meaning of the Diagram • A smooth bell-shaped curve. • Most data values are near the middle (mean). • Fewer values occur near the extremes (tails). • Symmetrical on both sides. • Simple Biological Example • If Hb levels or heights follow a normal distribution: • Most children have Hb near the average • Very few have extremely low or high Hb • The curve explains how values are spread
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  Summary of Notation(Very Simple Table) Symbol Meaning Example n Number of trials Children tested (e.g., n = 10) p Probability of success Infection chance (e.g., p = 0.4) λ (lambda) Average rate of rare events Average egg count = 3 μ (mu) Mean of continuous data Average Hb level σ (sigma) Standard deviation Spread of Hb levels
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  A Simple Tableto Remember If data is… Distribution Examples Two outcomes only Binomial Infected / Not infected Rare events counted Poisson Egg counts, number of cases Continuous & natural Normal Hb levels, heights, weights
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  Why You Cannot Do Research Without Probability Distributio ns •They tell you which statistical test to use. • They help you avoid wrong conclusions. • They explain how nature behaves (infection patterns, measurements, rare events). • They guide sampling, study design, and interpretation. • Distributions are like maps they → guide a researcher toward the correct statistical road.
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  Thank u