# stat 425 lecture1

**Topics:**Normal distribution, Multivariate normal distribution, Probability theory

**Pages:**6 (259 words)

**Published:**December 3, 2013

Examples for 01/15/2013

Spring 2013

Bivariate Normal Distribution:

1

f (x, y ) =

1− ρ 2

2 π σ1 σ 2

1

−

2 1− ρ 2

exp

(

)

2

x − µ1

σ

1

x −µ1

−2ρ

σ

1

y −µ 2 y −µ 2

+

σ2 σ2

2

,

− ∞ < x < ∞, − ∞ < y < ∞.

(a)

2

2

the marginal distributions of X and Y are N µ 1 , σ 1 and N µ 2 , σ 2 ,

respectively;

(b)

the correlation coefficient of X and Y is

independent if and only if

(c)

σ2

2

( x − µ 1 ), (1 − ρ 2 )σ 2 ;

σ1

the conditional distribution of X, given Y = y, is

N µ1 + ρ

(e)

ρ = 0;

the conditional distribution of Y, given X = x, is

Nµ2 + ρ

(d)

ρ XY = ρ, and X and Y are

σ1

( y − µ 2 ), (1 − ρ 2 )σ 12 .

σ2

a X + b Y is normally distributed with

mean

E(aX + bY) = a µ1 + b µ2

variance

and

2

2

Var ( a X + b Y ) = a 2 σ 1 + 2 a b ρ σ 1 σ 2 + b 2 σ 2 .

ρ = 0.0

ρ = 0.3

ρ = 0.6

ρ = 0.9

1.

A large class took two exams. Suppose the exam scores X (Exam 1) and Y (Exam 2) follow a bivariate normal distribution with

µ 1 = 70,

µ 2 = 60,

σ 1 = 10,

σ 2 = 15,

ρ = 0.6.

a)

A students is selected at random. What is the probability that his/her score on Exam 2 is over 75?

b)

Suppose you're told that a student got a 80 on Exam 1. What is the probability that his/her score on Exam 2 is over 75?

c)

Suppose you're told that a student got a 66 on Exam 1. What is the probability that his/her score on Exam 2 is over 75?

d)

Suppose you're told that a student got a 70 on Exam 2. What is the probability that his/her score on Exam 1 is over 80?

e)

A students is selected at random. What is the probability that the sum of his/her Exam 1 and Exam 2 scores is over 150?

f)

What proportion of students did better on Exam 1 than on Exam 2?

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