Further Statistics: Discrete Distributions Depth

Method

1. To prove a mean or variance formula, start from the defining sum (E(X) = sum x P(X=x) or similar) and manipulate it into a form that matches a known series - a geometric series sum_{n=0}^infty rⁿ = 1/(1-r) for |r|<1, or the exponential series e^x = sum xⁿ/n! - rather than trying to sum an unfamiliar series directly.To prove a mean or variance formula, start from the defining sum (E(X) = sum x P(X=x) or similar) and manipulate it into a form that matches a known series - a geometric series sum_{n=0}^infty r^n = 1/(1-r) for |r|<1, or the exponential series e^x = sum x^n/n! - rather than trying to sum an unfamiliar series directly.
2. For a geometric-distribution proof, differentiate the geometric series sum_{x=0}^infty q^x = 1/(1-q) term by term with respect to q to generate sum x q^{x-1}, then multiply by the missing constant to match E(X).For a geometric-distribution proof, differentiate the geometric series sum_{x=0}^infty q^x = 1/(1-q) term by term with respect to q to generate sum x q^{x-1}, then multiply by the missing constant to match E(X).
3. For a Poisson-distribution proof, it is often easier to find E(X(X-1)) first, since the x(x-1) factor cancels part of the x! in the denominator of the probability function, then use Var(X) = E(X(X-1)) + E(X) - [E(X)]².For a Poisson-distribution proof, it is often easier to find E(X(X-1)) first, since the x(x-1) factor cancels part of the x! in the denominator of the probability function, then use Var(X) = E(X(X-1)) + E(X) - [E(X)]^2.
4. To prove the memoryless property of a geometric distribution, write P(X>s+t | X>s) = P(X>s+t AND X>s)/P(X>s) = P(X>s+t)/P(X>s) (since X>s+t implies X>s), then use P(X>n) = (1-p)ⁿ and simplify the quotient of powers.To prove the memoryless property of a geometric distribution, write P(X>s+t | X>s) = P(X>s+t AND X>s)/P(X>s) = P(X>s+t)/P(X>s) (since X>s+t implies X>s), then use P(X>n) = (1-p)^n and simplify the quotient of powers.
5. To find the distribution of X+Y for independent discrete X and Y, list every pair of values (i, k-i) that sum to the target k, multiply P(X=i) by P(Y=k-i) for each pair, and add the products: P(X+Y=k) = sum_i P(X=i) P(Y=k-i).To find the distribution of X+Y for independent discrete X and Y, list every pair of values (i, k-i) that sum to the target k, multiply P(X=i) by P(Y=k-i) for each pair, and add the products: P(X+Y=k) = sum_i P(X=i) P(Y=k-i).
6. Check a derived formula by testing it against a value you already know: does the derived variance formula give the right number for a specific λ or p you can check by direct summation of a few terms of the original series?Check a derived formula by testing it against a value you already know: does the derived variance formula give the right number for a specific lambda or p you can check by direct summation of a few terms of the original series?

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