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Chapter: Probability Distributions· Mathematics & Statistics (Commerce) Part 2

If \(X \sim P(m)\) with \(P(X = 1) = P(X = 2)\) then find the mean and \(P(X = 2)\). Given \(e^{-2} = 0.1353\)Solution : Since \(P(X = 1) = P(X = 2)\)\(\therefore \dfrac{e^{\square}m^1}{1!} = \dfrac{e^{-m}m^2}{\square}\)\(\therefore m = \square\)\(\t…

Q.6 (C)(ii)

If \(X \sim P(m)\) with \(P(X = 1) = P(X = 2)\) then find the mean and \(P(X = 2)\). Given \(e^{-2} = 0.1353\)

Solution : Since \(P(X = 1) = P(X = 2)\)

\(\therefore \dfrac{e^{\square}m^1}{1!} = \dfrac{e^{-m}m^2}{\square}\)

\(\therefore m = \square\)

\(\therefore P(X = 2) = \dfrac{e^{-2}\cdot 2^2}{2!} = \square\)

1 answer

Best answer

Answer: the boxes are \(-m\); \(2!\); \(2\); \(0.2706\). Mean \(= m = 2\) and \(P(X = 2) = 0.2706\).

For a Poisson variable, \(P(X = x) = \dfrac{e^{-m}m^x}{x!}\).

\(P(X = 1) = P(X = 2) \Rightarrow \dfrac{e^{-m}m^1}{1!} = \dfrac{e^{-m}m^2}{2!} \Rightarrow m = \dfrac{m^2}{2} \Rightarrow m = 2\) (as \(m \gt 0\)).

Mean of a Poisson distribution \(= m = 2\).

\(P(X = 2) = \dfrac{e^{-2}\cdot 2^2}{2!} = 2e^{-2} = 2 \times 0.1353 = 0.2706\).

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