1. The following table shows the probability distribution of X , the number of t

Question

1.   The following table shows the probability distribution of X, the number of television sets per household in a large city.

x

0

1

2

3

4

p(x)

.10

.40

.25

.15

.10

(a.) Compute the following probabilities:

(i) P(1 < X 2)

(ii) P(3 (X^ 2) + 2 < 15)

(iii) P( (X)^(1/2) + 2.6 4)

(b) Compute the expected value E(X).

(c) Find the variance and hence the standard deviation of X.

(d) What fraction of the measurements lie between (both end points inclusive) one standard deviation of the mean?

x

0

1

2

3

4

p(x)

.10

.40

.25

.15

.10

Explanation / Answer

i) P(1<X<=2) =P(X=2) =0.25

ii) P(3X2+2<15) =P(3X2<13)=P(X2<4.33)= P(X<2.08)=P(X=0)+P(X=1)+P(X=2) =0.1+0.4+0.25=0.75

iii) P(X1/2+2.6>=4)=P(X1/2 >=1.4)=P(X>=1.42)=P(X>=1.96)=P(X=2)+P(X=3)+P(X=4) =0.25+0.15+0.1=0.50

b)

from above  expected value E(X) =1.75

c) variance =1.2875

std deviation =1.1347

d) P(mean -st deviation<X<mean +std deviation) =P(1.75-1.1347<X<1.75+1.1347)

=P(0.6153<X<2.8847)=P(X=1)+P(X=2) =0.4+0.25 =0.65

x p(x) xP(x) x2P(x) (x-)2 (x-)2P(x) 0 0.100 0.000 0.000 3.063 0.306 1 0.400 0.400 0.400 0.563 0.225 2 0.250 0.500 1.000 0.063 0.016 3 0.150 0.450 1.350 1.563 0.234 4 0.100 0.400 1.600 5.063 0.506 total 1 = 1.75 4.350 10.313 2= 1.2875 std deviation=     =    2 = 1.1347

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