Understanding Inferential Statistics Concepts I'll explain the key concepts from your PDF on inferential statistics, focusing on the two main methods presented: the cumulative distribution function (CDF) method and the moment generating function (MGF) method. 1. Cumulative Distribution Function (CDF) Method This method is used to find the probability distribution of a transformed random variable Y = g(X) when you know the distribution of X. Steps: Find the CDF of X: Fₓ(x) = P(X ≤ x) Express X in terms of Y: Find the inverse function g⁻¹(y) Find the CDF of Y: Fᵧ(y) = Fₓ(g⁻¹(y)) Differentiate to get the PDF of Y: fᵧ(y) = d/dy Fᵧ(y) Determine the range of Y Example 1: Given X with PDF: f(x) = { 1/2, 0 < x < 2; 0 elsewhere } Find PDF of Y = -X² Solution: Fₓ(x) = ∫(1/2)dt from 0 to x = x/2 Y = -X² ⇒ X = √(-Y) (since X > 0) Fᵧ(y) = Fₓ(√(-y)) = √(-y)/2 fᵧ(y) = d/dy [√(-y)/2] = -1/(4√(-y)) Range: When X∈(0,2), Y∈(-4,0) Example 2: Given X ~ U[-1,1], find PDF of Y = eˣ Solution: Fₓ(x) = (x+1)/2 for x∈[-1,1] Y = eˣ ⇒ X = ln(Y) Fᵧ(y) = P(Y≤y) = P(eˣ≤y) = P(X≤ln(y)) = Fₓ(ln(y)) = (ln(y)+1)/2 fᵧ(y) = d/dy [(ln(y)+1)/2] = 1/(2y) Range: When X∈[-1,1], Y∈[e⁻¹,e¹] 2. Moment Generating Function (MGF) Method The MGF of a random variable X is defined as: Mₓ(t) = E[eᵗˣ] For discrete X: Mₓ(t) = Σ eᵗˣⁱ fₓ(xᵢ) For continuous X: Mₓ(t) = ∫ eᵗˣ fₓ(x) dx Properties: If Y = aX + b, then Mᵧ(t) = eᵇᵗMₓ(at) MGF uniquely determines the distribution (if it exists in a neighborhood of 0) Example 3: Find MGF of exponential distribution with parameter λ=2 Solution: f(x) = 2e⁻²ˣ for x≥0 Mₓ(t) = ∫₀^∞ eᵗˣ 2e⁻²ˣ dx = 2∫₀^∞ e⁻⁽²⁻ᵗ⁾ˣ dx = 2/(2-t) for t<2 Example 4: Find MGF of normal distribution N(μ,σ²) Solution: Mₓ(t) = exp(μt + σ²t²/2) Example 5: Given X ~ N(μ,σ²), find distribution of Y = (X-μ)/σ Using MGF method: Mᵧ(t) = E[eᵗʸ] = E[eᵗ⁽ˣ⁻μ⁾/σ] = e⁻μᵗ/σ Mₓ(t/σ) = e⁻μᵗ/σ exp(μ(t/σ) + σ²(t/σ)²/2) = exp(t²/2) This is MGF of standard normal N(0,1) Example 6: Given X ~ N(μ,σ²), find distribution of Y = a + bX Using MGF method: Mᵧ(t) = eᵃᵗMₓ(bt) = eᵃᵗ exp(μbt + σ²b²t²/2) = exp((a + μb)t + (σb)²t²/2) This is MGF of N(a + μb, (σb)²) Key Takeaways: CDF Method is useful for finding distributions of transformed variables by: Working through the CDF Using the transformation relationship Differentiating to get the PDF MGF Method is powerful because: MGFs uniquely characterize distributions Linear transformations of normal variables remain normal Standardizing a normal variable (subtracting mean, dividing by SD) gives standard normal Normal Distribution Properties: Linear transformations preserve normality Standard normal (Z = (X-μ)/σ) has mean 0 and variance 1 MGF of normal is exp(μt + σ²t²/2)