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Statistics 2 - Coggle Diagram
Statistics 2
Common distributions
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Binomial Random variable
X ~ Bernoulli (n, p) where n = no. trials, p is the probability of success
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PMF = P(X = k) = C(n, k) p^k (1 - p)^(n - k) for k = 0, 1, 2, ..., n
Validity of PMF = sum of C(n, k) p^k * (1-p)^(n-k) =1
model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes, p or 1-p
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Geometric Distribution
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Range = {1,2,3.....n} some people even start it from 0
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negative binomial
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Range = { r r+1, r+2 .......}
PMF = C ((k-1), (r-1)) * (1-p)^(k-r) p^r (if r = 1 then it is similar to a geometric distribution
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Poisson Random Variable
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Validity of PMF : check using the expansion of e^x, e^(-λ) = 1 + λ + λ
^2 / 2! +......
as λ increases, the graph shifts towards right
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chi distribution
with n degrees of freedom, chi distribution = gamma (n/2, 1/2)
Normal distribution
square of Normal(0, sigma^2) is gamma (1/2, 1/(2sigma^2)
Exponential
Sum of n iid exp(x) distributions = gamma (n, x)
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