Cambodia

STA 2001 Basic Concepts in Statistics and Probability I 29/11/2006 Tutorial 11 • Normal Distribution X ∼ N (µ, σ 2 ): - p.d.f.: f (x) =

(x−µ2 ) 1 √ exp(− 2σ 2 ), σ 2π 2 σ 2 t2 2 ).

for −∞ < x < ∞.

- E(X) = µ, V ar(X) = σ . - M (t) = exp(µt +

- Standard normal distribution:N (0, 1). • Random functions associated with Normal Distributions. If X1 , X2 , . . . , Xn is a random sample from the normal distribution N (µ, σ 2 ), ¯ - Sample mean X = - sample variance S 2 =

n i=1

Xi

n

n

∼ N (µ, σ ) n ∼ χ2 (n − 1)

2

i=1

¯ (Xi −X) (n−1)S 2 , σ2 n−1

• Sum of independent Normals. If X1 , X2 , . . . , Xn are mutually independent normal random variables with means µ1 , . . . , µn and variances n n n 2 2 σ1 , ..., σn respectively. Then Y = i=1 ci Xi follows the normal distribution N ( i=1 ci µi , i=1 c2 µ2 ) i i • Central Limit Theorem. ¯ If X is the mean of a random sample X1 , . . . , Xn from a distribution with finite mean µ and a finite positive σ 2 , then x−µ ¯ W = σ/√n ∼ N (0, 1), as n → ∞ Examples Example1: Let X denote the wing length in millimeters of a male gallinule and Y the wing length in millimeters of a female gallinule. Assume that X is N(184.09, 39.37) and Y is N(171.93, 50.88), and that X and Y are independent. If a male and a female gallinule are captured, what is the probability that X is greater than Y?

Example2: Let X equal the birth weight in grams of a baby born in the Sudan. Assume that E(X) = 3320 and V ar(X) = 6602. ¯ ¯ Let X be the sample mean of a random sample of size n = 225. Find P (3233.76 ≤ X ≤ 3406.24), approximately.

Example3: √ X and Y are independent, with the same distribution, what is the p.d.f. of Z = X 2 + Y 2 ?

Example4: If Z is standard normal, then P (|Z| ≥ t) ≤

2 e−t π t

2 /2

, for all t > 0