Answer :

(i) Given :

To find : mean (π), variance (Ο^{2}) and standard deviation (Ο)

Formula used :

Mean = E(X) =

Variance = E(X^{2}) -

Standard deviation =

Mean = E(X) = = x_{1}P(x_{1}) + x_{2}P(x_{2}) + x_{3}P(x_{3}) + x_{4}P(x_{4})

Mean = E(X) = 0() + 1() +2() +3() = 0 + + + = = =

Mean = E(X) = = 1.2

= = 1.44

E(X^{2}) = = P(x_{1}) + P(x_{2}) + P(x_{3}) + P(x_{4})

E(X^{2}) = () + () + () + () = 0 + + + = =

E(X^{2}) = 2

Variance = E(X^{2}) - = 2 β 1.44 = 0.56

Variance = E(X^{2}) - = 0.56

Standard deviation = = = 0.74

Mean = 1.2

Variance = 0.56

Standard deviation = 0.74

(ii) Given :

To find : mean (π), variance (Ο^{2}) and standard deviation (Ο)

Formula used :

Mean = E(X) =

Variance = E(X^{2}) -

Standard deviation =

Mean = E(X) = = x_{1}P(x_{1}) + x_{2}P(x_{2}) + x_{3}P(x_{3}) + x_{4}P(x_{4})

Mean = E(X) = 1(0.4) + 2(0.3) +3(0.2) +4(0.1) = 0.4 + 0.6 + 0.6 + 0.4 = 2

Mean = E(X) = 2

= = 4

E(X^{2}) = = P(x_{1}) + P(x_{2}) + P(x_{3}) + P(x_{4})

E(X^{2}) = (0.4) + (0.3) + (0.2) + (0.1) = 0.4 + 1.2 + 1.8 + 1.6 = 5

E(X^{2}) = 5

Variance = E(X^{2}) - = 5 β 4 = 1

Variance = E(X^{2}) - = 1

Standard deviation = = = 1

Mean = 2

Variance = 1

Standard deviation = 1

(iii) Given :

To find : mean (π), variance (Ο^{2}) and standard deviation (Ο)

Formula used :

Mean = E(X) =

Variance = E(X^{2}) -

Standard deviation =

Mean = E(X) = = x_{1}P(x_{1}) + x_{2}P(x_{2}) + x_{3}P(x_{3}) + x_{4}P(x_{4})

Mean = E(X) = -3(0.2) + (-1)(0.4) + 0(0.3) + 2(0.1)= -0.6 - 0.4 + 0 + 0.2 = -0.8

Mean = E(X) = -0.8

= = 0.64

E(X^{2}) = = P(x_{1}) + P(x_{2}) + P(x_{3}) + P(x_{4})

E(X^{2})= (0.2) + (0.4) + (0.3) + (0.1) = 1.8 + 0.4 + 0+ 0.4 = 2.6

E(X^{2}) = 2.6

Variance = E(X^{2}) - = 2.6 β 0.64 = 1.96

Variance = E(X^{2}) - = 1.96

Standard deviation = = = 1.4

Mean = -0.8

Variance = 1.96

Standard deviation = 1.4

(iv) Given :

To find : mean (π), variance (Ο^{2}) and standard deviation (Ο)

Formula used :

Mean = E(X) =

Variance = E(X^{2}) -

Standard deviation =

Mean = E(X) = = x_{1}P(x_{1}) + x_{2}P(x_{2}) + x_{3}P(x_{3}) + x_{4}P(x_{4}) + x_{5}P(x_{5})

Mean = E(X) = -2(0.1) + (-1)(0.2) + 0(0.4) + 1(0.2) + 2(0.1)

Mean = E(X) = -0.2 - 0.2 + 0 + 0.2 + 0.2 = 0

Mean = E(X) = 0

= = 0

E(X^{2}) = = P(x_{1}) + P(x_{2}) + P(x_{3}) + P(x_{4}) + P(x_{5})

E(X^{2}) = (0.1) + (0.2) + (0.4) + (0.2) + (0.1)

E(X^{2}) = 0.4 + 0.2 + 0 + 0.2 +0.4 = 1.2

E(X^{2}) = 1.2

Variance = E(X^{2}) - = 1.2 β 0 = 1.2

Variance = E(X^{2}) - = 1.2

Standard deviation = = = 1.095

Mean = 0

Variance = 1.2

Standard deviation = 1.095

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