stat 425 lecture1
STAT 420
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
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