University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

113

Pages:

3
Author:

Niklas

Equations: P(Z > -0.5) = 1 - P(Z < -0.5) = 0.6915 X ~ N(4, 16) P(x0.15 ≤ X ≤ x0.55) = F(x0.55) - F(x0.15) P(7 ≤ X ≤ 25) = 0.67 Zα = -1.64 Convergence in Probability: A sequence of random variables Xn converges in probability to X if and only if lim(n→∞) P(|Xn - X| > ε) = 0 for all ε > 0. Each trajectory is a realization of a sequence of random variables. The idea is that as n increases, the results concentrate around the limit of the sequence. Xi = 1 if head, 0 otherwise The mean Xn is the proportion of heads: E(Xn) = P(head) Given a sequence of independent random variables X1, X2, ..., Xn with the same mean E(Xi) = μ and the same finite variance V(Xi) = σ², the sample mean is: X̄n = (1/n) * Σi=1^n Xi As n increases, the probability that X̄n is close to μ approaches 1. P(|X̄n - μ| ≤ ε) → 1 for any ε > 0. Note that there is also a strong law of large numbers that involves a different type of convergence. Example: From an urn half-filled with white balls, draw n=10 balls with replacement. The expected proportion of white balls is p=0.5. The probability of extracting exactly p=0.5 white balls is not 1. We can only compute the probability to observe this outcome. The probability of observing proportions of white balls within a certain range (e.g., 0.4 to 0.6) increases with n. Consider a random variable S following a Binomial distribution with dimension n and probability p. It converges in distribution to a Normal distribution with mean np and variance np(1-p). If n is small, then you should use the binomial distribution. If n is large, you should use the normal distribution. If S ~ Binom(n, p), with n large enough, then approximately: P(S ≤ x) ≈ Φ((x - np) / √(np(1-p))) We can use the normal distribution to compute probabilities related to a Binomial. Example: An airline expects reservations for a plane with 200 seats, knowing that 12% of reservations are no-shows. X ~ Binom(200, 0.9) represents the number of passengers on the plane. P(X = 200) ≈ Φ((200 - 180) / √(180 * 0.1)) = 0.0398 P(190 ≤ X ≤ 210) ≈ Φ((210 - 180) / √(180 * 0.1)) - Φ((190 - 180) / √(180 * 0.1)) = 0.9318