# 2016_10_11 — Seminar 6 — Independence, Binomial

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Seminar 6. Independence and Binomial random variables

1.
Are the following random variables independent?

2.

(A) Find the joint distribution of X, Y.

(B) Find E(XY), Var
(X+Y), Var(X
-
Y)

(C) Find E(X+Y)
2
, E(X(X+Y)
2
)

(
D
) Find P(X=1, Y1), P(X-4,; P; X50;1,Y-4,; P; X50;
-
1)

(
E
) Find E(X+Y | X=2), E(XY | Y=
-
1)

3.

A biased coin lands heads
-
up with probability 0.7. Suppose this coin is tossed 100 times.

(A) What is the expected number of heads and it
s variance?

(C) Find the probability that the number of heads differs from its expected value by no more than 3.

4.

Company “O’Bender” has got a cont
ract to deliver 200 chairs. During transportation a chair may be damaged with
probability 0.1.

(A) To be sure to meet the contract, “O’Bender” plans to ship 225 chairs. What is the probability the terms of the contract

(B) How many chairs must

be shipped to be 99% sure to meet the terms of the contract?

(C) If it is possible to transport chairs more accurately, what should be the maximal probability to damage a chair in order
to meet the terms of the contract with probability 99% when transport
ing 225 chairs.

5
.

A multiple
-
choice test consists of 50 questions. Suppose a students knows answers to 10 of them and will try to guess
answers to the other questions. Each question has 4 choices of answer and only 1 is correct.

(A) Find the expected
number and the standard deviation of correct answers given by the student.

(C) The parents of the student promise him to buy a new iPad i
f he passes the test. Assuming the price of an iPad is \$800,
find the expectation and standard deviation of the parents' expenses.

X
\
Y

1

2

1

0.1

0.2

2

0.3

0.4

X
\
Y

1

2

3

4

0.1

0.2

0.1

5

0.3

0

0.3

X
\
Y

0

2

3

-
1

0

0.1

0.1

1

0.1

0.4

0.1

2

0.1

0.1

0

X
\
Y

1

2

3

4

1

0.12

0.03

0.06

0.09

2

0.2

0.05

0.1

0.15

3

0.08

0.02

0.04

0.06

X

1

2

3

P

0.3

0.2

0.5

Y

-
1

0

1

P

0.4

0.4

0.2

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