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  • A class has 15 students whose ages are 14,17,15,14, 21,17,19,20,16,18, 20,17,16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X,

Q. 14  A class has 15 students whose ages are 14,17,15,14,21,17,19,20,16,18,20,17,16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.

Answers (1)

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Total students = 15

probability of selecting a student :

                                                          =\frac{1}{15}

The information given can be represented as frequency table :

X 14 15 16 17 18 19 20

21
f 2 1 2 3 1 2 3 1

P(X=14)=\frac{2}{15}                                          P(X=15)=\frac{1}{15}                                           P(X=16)=\frac{2}{15}

P(X=17)=\frac{3}{15}=\frac{1}{5}                              P(X=18)=\frac{1}{15}                                            P(X=19)=\frac{2}{15}

P(X=20)=\frac{3}{15}=\frac{1}{5}                                P(X=21)=\frac{1}{15}

Probability distribution is as :

X 14 15 16 17 18 19 20 21
P(X) \frac{2}{15} \frac{1}{15} \frac{2}{15} \frac{1}{5} \frac{1}{15} \frac{2}{15} \frac{1}{5} \frac{1}{15}

E(X)=14\times \frac{2}{15}+15\times \frac{1}{15}+16\times \frac{2}{15}+17\times \frac{1}{5}+18\times \frac{1}{15}+19\times \frac{2}{15}+20\times \frac{1}{5}+21\times \frac{1}{15}

E(X)=\frac{263}{15}=17.53

E(X^2)=14^2\times \frac{2}{15}+15^2\times \frac{1}{15}+16^2\times \frac{2}{15}+17^2\times \frac{1}{5}+18^2\times \frac{1}{15}+19^2\times \frac{2}{15}+20^2\times \frac{1}{5}+21^2\times \frac{1}{15}

E(X^2)=\frac{4683}{15}=312.2

Variance =E(X^2)-(E(X))^2

Variance =312.2-(17.53)^2

Variance =312.2-307.42

Variance =4.78

Standard\, \, deviation =\sqrt{4.78}=2.19

Posted by

seema garhwal

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