Question

For a Poission distribution which pair has same value.

• A. (Mean, Std. Deviation)
• B. (Variance, Standard Deviation)
• C. (Mean, Variance)
• D. None of these

Answer 1

The correct option is C (Mean, Variance)

For Poission distribution we have $P\left(r\right)=\frac{e^{-\pi }{\lambda }^r}{r^i};(r=0,1,2,...\infty )$

mean $=\frac{\sum f_i{x}_i}{\sum f_i}=\sum _{r=0}^{\infty }rP\left(r\right),\sum f_{i}P\left(r\right)=1$$=0+\lambda e^{-\lambda }+2\frac{{\lambda }^2{e}^{-\lambda }}{2!}+3\frac{{\lambda }^3{e}^{-\lambda }}{3!}+.....\infty$ $=\lambda e^{-\lambda }(1+\frac{\lambda }{11}+\frac{{\lambda }^2}{2!}+\frac{{\lambda }^3}{3!}+.....\infty )$$=\lambda e^{-\lambda }.e^{\lambda }=\lambda$

Similarly, $o^2=$ Variance $={\sum }_{r=0}^{\infty }{r}^{2}P\left(r\right)-{\left({\sum }_{r=0}^{\infty }rP\left(r\right)\right)}^2$ $=\lambda e^{\lambda }(e^{-\lambda }+\lambda e^{-\lambda })-{\lambda }^2=\lambda =$ mean

$\therefore$ mean$=$variance