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Polytechnic Institute of NYUMA 2212 MIDTERM SAMPLE

Print Name:

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Section:

ID #:

Directions: You MUST SHOW ALL YOUR WORK as neatly and clearly

as possible and indicate the final answer clearly. You may use a calculator.

Problem Possible Points

1 15

2 15

3 20

4 15

5 20

5 15

Total 100

1

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(1) (15 points) Telephone calls arrive a college switchboard at an average rate of 3 callsevery 4 minutes according to a Poisson process.

(a) Find the probability that there will be at least 3 calls in the next 12 minutes.

(b) Find the probability that there will be no calls in the next 12 minutes.

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(2) (15 points) A committee of 16 persons is selected randomly from a group of 400people, of whom are 240 are women and 160 are men. Approximate the probabilitythat the committe contains at least 3 women.

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(3) (20 points) The probability mass function (p.m.f.) of some random variable X is

f(0) = 0.4f(x) = cx ifx= 1, 2, 3

(a) Determine c.(b) What are the mean, variance and standard deviation ofX?

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(4) (15 points) The 2% of certain candy is underweight, and you buy 60 candies. Ap-proximate the probability that at most 4 candies are underweight among the 60.

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(5) (20 points) For a random variable X, the mean is E(X) = 0.6, and the probabilitymass function (p.m.f.) of is a +bx ifx= 0, 1.(a) Determine a, b.(b) What are the variance and the standard deviation ofX?

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(6) (15 points) The results of rolling a six-sided dice five times are x1 = 4, x2 = 1,x3 = 3, x4 = 6, x5= 3.

(a) Determine the sample mean, sample variance and sample standard deviation.

(b) Based on the answers in (a), may the data be approximated by a Poissondistribution?

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Some definitions and formulas you might find useful

(1) For a random sample x1, x2, . . . , xn, the sample mean and the sample variance aredefined as

x= 1n

ni=1

xi and s2 = 1n 1

ni=1

(xi x)2.

And the sample standard deviation is simply s=

s2.

(2) P(n, r) = n!

(n r)! ; C(n, r) =

n

r

=

n!

r!(n r)! .

(3) IfAand B are any two events, then

P(A B) =P(A) +P(B) P(A B).

(4) The conditional probability ofA, given B, is P(A|B) = P(A

B)

P(B) , ifP(B)> 0.

(5) Events Aand B are independent if and only ifP(A B) =P(A)P(B). OtherwiseAand B are called dependent events.

(6) (Bayes Theorem) If the events B1, B2, . . ., Bk constitute a partition of the samplespace S, where P(Bi) > 0 for i = 1, 2, . . . , k, then for any event A in S such thatP(A)> 0,

P(Br|A) = P(Br A)ki=1

P(Bi

A)=

P(Br)P(A|Br)ki

=1

P(Bi)P(A|Bi)

for r= 1, 2, . . . , k .

(7) Iff(x) is the p.m.f. of a discrete random variable X, then

= E(X) =

xpossible value

xf(x)

is the expected value, or mean ofX. In addition

E(X2) =

xpossible value

x2f(x)

is the second moment, and

Var (X) =E[(X

)2] =E(X2)

2

is the variance, and the standard deviation ofXis defined to be = Var X. Themoment generating function of the Xabove is

M(t) =

xpossible value

etxf(x)

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A table for some well-known distributions. (Note: In the following table q= 1 p.)

Name p.m.f m.g.f mean variance

Binomial(n, p) f(x) =

n

x

pxqnx M(t) = (q+pet)n np npq

x= 0, 1, 2, . . . , n

Geometric(p) f(x) =qx1p M(t) = pet

1 qet1

p

q

p2

x= 1, 2, . . .

Negative f(x) =

x 1r 1

prqxr M(t) = (pe

t)r

(1 qet)rr

prqp2

Binomial (r, p)x= r, r+ 1, r+ 2, . . .

Poisson() f(x) =ex

x! M(t) =e(e

t1)

x= 0, 1, 2, . . .Hypergeometric

f(x) =N1

x N2

nxNn

nN1N

(N1, N2, N=N1+N2, n) x= 0, 1, . . . , min{n, N1}

For Geometric distributionwith probability p, the probability that the first k trials arefailures is (1 p)k.

Approximation by Poisson distribution (Using the notation of the Table above)

Name Approximation

Binomial(n, p)

n

x

pxqnx e

x

x! = np

Hypergeometric

N1

x

N2

nx

N

n

exx!

= nN1N