stats quiz 5 ans

7/27/2019 Stats Quiz 5 Ans

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Answers Quiz 5

Probability and StatisticsMarcia Schafgans

September 2013

1. Consider 100 independent random variables uniformly distributed in the [0; 1] interval. Call each

of them Xi. Let Y =P100

i=1 Xi. Then, we can say that the distribution of probability of Y is:

(a) aproximately normal with mean exactly equal to 50.

(b) aproximately normal with mean close (but probably not equal) to 50.

(c) uniform in the [0; 100] interval.

(d) fY(y) =

y=2500 for 0 y 50

(100 y)=2500 for 50 y 100(e) none of the above.

CLT =) Y NormalE(Y) = E

P100i=1 Xi

=P100

i=1 E(Xi) =P100

i=11

2= 50.

ANSWER: a

2. (LM) What is the MLE for in the pdf

f(y; ) = 2y1 2 ; y 1

if a random sample of size 6 yielded the measurements 0.70, 0.63, 0.92, 0.86, 0.43 and 0.21?

(a) 0

(b) 0:094

(c) 0:18

(d) 0:21

(e) 0:625

L() =

8 2:7) = 0:47 (and clearly Ymax is always 3).

ANSWER: b

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5. (LM) What is the method of moments estimate of if a random sample of size 5 consisting of thenumbers 17, 92, 46, 39 and 56 is drawn from the pdf:

f(y; ) =1

; 0 y

(a) 92

(b) 100

(c) 104

(d) 109

(e) 112

Mean of the distribution: E(Y) = =2: MME chooses such that 1nyi = =2Sample mean: y = 50

MME2

= y = 50 =) MME = 100

ANSWER: b

6. Consider a random variable X N(; 2), both and 2 are unknown. A researcher was tryingto estimate and from a sample of size 9, obtained x = 1 and s = 3. Which one of the followingis a 95% condence interval for ?

(a) (1; 2:86](b) [5:91; 7:91](c) [4:58; 6:58](d) [0:86; 2:86](e) [1:31;1)

x = 1, s=p

n = 1t8;5% = 1:86, t8

;2:5% = 2:306

(a) Pr(x > 2:86) = Pr(t8 > 1:86) = 5%, correct

ANSWER: a

7. Let Y1; Y2;:::;Yn be a random sample of size n from the pdf f(y; ) =1

ey= ; > 0; y 0.

Consider 2 estimators: 1 = Y1 and 2 = n:Ymin.Hint: the pdf of Ymin is given by:

fYmin(y) = n:fY(y): [1 FY(y)]n1

We can say that:

(a) both 1 and 2 are biased

(b) one of the estimators (1 or 2) is biased, the other is not.

(c) both 1 and 2 are unbiased and 1 is more ecient.

(d) both 1 and 2 are unbiased and 1 and 2 are equally ecient.

(e) both 1 and 2 are unbiased and 2 is more ecient.

FY(y) = 1 ey= . So:

fYmin

(y) = n1

ey= :

ey=

n1=

n

e

n

y

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which is the pdf of the exponential distribution with mean =n.So,1: mean is , standard deviation is .n: mean is n=n = , standard deviation is n=n = .

ANSWER: d

8. Let X be a normally distributed random variable (X N(; 2)). A random sample of 3 real-izations of X is generated and the sample mean, X, is calculated. Then, another realization of Xis collected (call it X4, it is independent from the 3 previous ones). What is the probability thatX4 2 [ X 2; X+ 2]?

(a) 68%

(b) 84%

(c) 91%

(d) 95%

(e) 98%

X N(; 2

3)

SoE(X4 X) = 0

and as X and X are independent,

V ar(X4 X) = 2 +2

3=

42

3

Stdv(X4 X) =2p

3

Now:

Pr(X4 2 [ X 2; X+ 2])= Pr(X4 X2 [2; 2])

= Pr

X4 X

2p3

2"22p3

;22p3

#!

= Pr

Z2hp

3;p

3i

where Z has a standard normal distribution.Asp

3 = 1:7 and from the table we get (1:7) = 0:9554, 1 (1:7) = 0:044_6So, Pr Z2

p3;p

3 = 0:9554 0:044 _6 = 0:9108ANSWER: c

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stats quiz 5 ans

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