First slide

Random variable and probability distribution

Question

In a poisson distribution, the variance is m. The sum of terms in odd places in the distribution is

Moderate

A

$e^{- m}$
B

$e^{- m} \cosh (m)$
C

$e^{- m} \sinh (m)$
D

$e^{- m} \coth (m)$

Solution

$λ = m$
Sum of Odd terms $= P (X = 0) + P (X = 2) + P (X = 4) + ...........$

$= e^{- m} [1 + \frac{m^{2}}{2!} + \frac{m^{4}}{4!} + ..........]$

$= \frac{e^{- m}}{2} [[1 + \frac{m}{1!} + \frac{m^{2}}{2!} + \frac{m^{3}}{3!} + ...........] + [1 - \frac{m}{1!} + \frac{m^{2}}{2!} - \frac{m^{3}}{3!} + ...........]]$

$= e^{- m} [\frac{e^{m} + e^{- m}}{2}] = e^{- m} \cosh (m)$

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