Statistics: Distributions Depth

Method

1. Decide what the question needs. A new variable defined as Y = aX + b (or built from several independent trials): use E(aX+b) = aE(X) + b and Var(aX+b) = a² Var(X). One unknown parameter (μ or σ) of a normal distribution with one given probability: standardise and use the inverse normal function. BOTH μ and σ unknown, with two given probabilities: form two equations and solve simultaneously. Approximating a binomial with a normal distribution: check np>5 and n(1-p)>5, then apply a continuity correction.Decide what the question needs. A new variable defined as Y = aX + b (or built from several independent trials): use E(aX+b) = aE(X) + b and Var(aX+b) = a^2 Var(X). One unknown parameter (mu or sigma) of a normal distribution with one given probability: standardise and use the inverse normal function. BOTH mu and sigma unknown, with two given probabilities: form two equations and solve simultaneously. Approximating a binomial with a normal distribution: check np>5 and n(1-p)>5, then apply a continuity correction.
2. For X ~ B(n,p), E(X) = np and Var(X) = np(1-p). For any linear transformation Y = aX + b, E(Y) = aE(X) + b and Var(Y) = a² Var(X); adding a constant b never changes the variance, only multiplying by a does.For X ~ B(n,p), E(X) = np and Var(X) = np(1-p). For any linear transformation Y = aX + b, E(Y) = aE(X) + b and Var(Y) = a^2 Var(X); adding a constant b never changes the variance, only multiplying by a does.
3. To find one unknown parameter from a single probability statement about X ~ N(μ, σ^2): standardise using z = (x-μ)/σ, find the exact z-value that gives the stated probability using the inverse normal function, then substitute and rearrange for the unknown.To find one unknown parameter from a single probability statement about X ~ N(mu, sigma^2): standardise using z = (x-mu)/sigma, find the exact z-value that gives the stated probability using the inverse normal function, then substitute and rearrange for the unknown.
4. When BOTH μ and σ are unknown, form two equations of the form (x - μ)/σ = z from two different given probabilities (each with its own x-value and its own z-value), then solve simultaneously, typically by subtracting one equation from the other to eliminate μ first.When BOTH mu and sigma are unknown, form two equations of the form (x - mu)/sigma = z from two different given probabilities (each with its own x-value and its own z-value), then solve simultaneously, typically by subtracting one equation from the other to eliminate mu first.
5. For a normal approximation Y ~ N(np, np(1-p)) to X ~ B(n,p), apply a continuity correction that always widens a strict inequality into the equivalent range for the continuous variable: P(X > r) becomes P(Y > r+0.5); P(X ≥ r) becomes P(Y > r-0.5); P(X < r) becomes P(Y < r-0.5); P(X ≤ r) becomes P(Y < r+0.5).For a normal approximation Y ~ N(np, np(1-p)) to X ~ B(n,p), apply a continuity correction that always widens a strict inequality into the equivalent range for the continuous variable: P(X > r) becomes P(Y > r+0.5); P(X >= r) becomes P(Y > r-0.5); P(X < r) becomes P(Y < r-0.5); P(X <= r) becomes P(Y < r+0.5).
6. Always check np > 5 and n(1-p) > 5 before using a normal approximation to a binomial distribution; if either condition fails, use the exact binomial distribution instead.

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