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4 Problems on Random Variables - Assignment 1 | STAT 704, Assignments of Statistics

University of South Carolina (USC) - ColumbiaStatistics
Prof. D. Hitchcock

Material Type: Assignment; Professor: Hitchcock; Class: DATA ANALYSIS I; Subject: Statistics; University: University of South Carolina - Columbia; Term: Fall 2009;

Typology: Assignments

Pre 2010

Uploaded on 10/01/2009

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Homework 1 – STAT 704
1. Let Y11,...,Y1n1be a sample from a population with mean µ1and variance σ2
1. Let
Y21,...,Y2n2be a sample from another population with mean µ2and variance σ2
2.
Define
¯
Y1=
n1
X
j=1
Y1j
n1
,¯
Y2=
n2
X
j=1
Y2j
n2
.
(a) Find E(¯
Y1−¯
Y2).
(b) If ¯
Y1and ¯
Y2are independent, find var(¯
Y1−¯
Y2).
(c) If the two populations are normal (and ¯
Y1and ¯
Y2are independent), then does
¯
Y1−¯
Y2have a normal distribution? Explain why or why not.
2. Suppose Y1, Y2,...,Ynare independent random variables with mean µand variance σ2.
(a) Show that
(n−1)S2=
n
X
i=1
Y2
i
−n¯
Y2.
(b) Show that E(S2) = σ2. (Hint: Use the fact that var(Y) = E(Y2)−[E(Y)]2].)
3. Let Y1, Y2, Y3be independent random variables with means µ1, µ2, µ3and a common
variance σ2. Define
¯
Y=1
3
3
X
i=1
Yi.
(a) Find the covariance between Y1−¯
Yand ¯
Y.
(b) Find the expected value of (Y1+ 2Y2−Y3)2.
4. Let Y1and Y2be random variables with expected values µ1and µ2and variances σ2
1
and σ2
2.
(a) Show that cov(Y1+Y2, Y1−Y2) = σ2
1
−σ2
2.
(b) If W=Y1+Y2and V=Y1−Y2, then under what condition(s) can we be assured
that Wand Vare independent random variables?
1

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Download 4 Problems on Random Variables - Assignment 1 | STAT 704 and more Assignments Statistics in PDF only on Docsity!

Homework 1 – STAT 704

  1. Let Y 11

,... , Y

1 n 1

be a sample from a population with mean μ 1

and variance σ

2

1

. Let

Y

21

,... , Y

2 n 2

be a sample from another population with mean μ 2

and variance σ

2

2

Define

Y

1

n 1 ∑

j=

Y 1 j

n 1

Y

2

n 2 ∑

j=

Y 2 j

n 2

(a) Find E(

Y

1

Y

2

(b) If

Y

1

and

Y

2

are independent, find var(

Y

1

Y

2

(c) If the two populations are normal (and

Y

1

and

Y

2

are independent), then does

Y

1

Y

2

have a normal distribution? Explain why or why not.

  1. Suppose Y 1 , Y 2 ,... , Yn are independent random variables with mean μ and variance σ

2

.

(a) Show that

(n − 1)S

2

=

n ∑

i=

Y

2

i

− n

Y

2

.

(b) Show that E(S

2

) = σ

2

. (Hint: Use the fact that var(Y ) = E(Y

2

) − [E(Y )]

2

].)

  1. Let Y 1

, Y

2

, Y

3

be independent random variables with means μ 1

, μ 2

, μ 3

and a common

variance σ

2

. Define

Y =

3 ∑

i=

Yi.

(a) Find the covariance between Y 1

Y and

Y.

(b) Find the expected value of (Y 1

+ 2Y

2

− Y

3

2

.

  1. Let Y 1

and Y 2

be random variables with expected values μ 1

and μ 2

and variances σ

2

1

and σ

2

2

(a) Show that cov(Y 1

+ Y

2

, Y

1

− Y

2

) = σ

2

1

− σ

2

2

(b) If W = Y 1

+ Y

2

and V = Y 1

− Y

2

, then under what condition(s) can we be assured

that W and V are independent random variables?