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  • If the variables take the values with frequencies <img alt="\mathrm{{ }^{

If the variables take the values 0,1,2,3, \ldots, 10 with frequencies \mathrm{{ }^{10} C_0,{ }^{10} C_1,{ }^{10} C_2, \ldots,{ }^{10} C_{10}} respectively, then the variables of the distribution is
 

Option: 1

10


Option: 2

\sqrt10


Option: 3

5


Option: 4

\frac{5}{2}


Answers (1)

best_answer

\mathrm{\bar{x}=\frac{\sum_{r=0}^{10} r^{10} C_r}{\sum_{r=0}^{10}{ }^{10} C_r}=\frac{10 \cdot 2^9}{2^{10}}=5 \quad\left[\because \sum_{r=0}^n r \cdot{ }^n C_r=n \cdot 2^{n-1}\right]}

Also,\mathrm{ \sigma^2=\frac{\sum_{r=0}^{10} r^2 \cdot{ }^{10} C_r}{\sum_{r=0}^{10}{ }^{10} C_r}-(\bar{x})^2 }

             \mathrm{=\frac{10(10+1) 2^{10-2}}{2^{10}}-(5)^2 }

                              \mathrm{ {\left[\because \sum_{r=0}^n r^2 \cdot{ }^n C_r=n(n+1) 2^{n-2}\right] } }

            \mathrm{=\frac{10 \times 11 \times 2^5}{2^{10}}-25 }

            \mathrm{=\frac{55}{2}-25=\frac{5}{2} }

Hence option 4 is correct.





 

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