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Q)

Alpha particles are emitted by a radio active source at an average rate of $5$ in a $20$ minutes interval.Using poisson distribution find the probability that there will be $2$ emission.$[e^{-5}=0.0067].$

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A)
Toolbox:
  • A random variable $X$ is said to have a poisson distribution if the probability mass function of $X$ is
  • $P(X=x)=\large\frac{e^{\Large -\lambda }\lambda^x}{x!}$$\qquad (x=0,1,2........$ for some $\lambda > 0)$
  • Constants of a poisson distribution :
  • Mean=Variance=$\lambda$
  • The parameter of the Poisson distribution is $\lambda$
  • A Poisson random variable corresponds to rare events.
Step 1:
Let $X$ be the random variable denoting the number of alpha particles emitted in a 20 minutes interval .
$X$ follows a poisson distribution with mean =5 particles in a 20 minutes interval.
$\therefore \lambda=5$
$X\sim P(5)$
$P(X=x)=\large\frac{e^{-5}5^x}{x!}$$\quad x=0,1,2.........$
Step 2:
Probability of 2 emissions in the 20 minutes interval is
$P(X=2)=\large\frac{e^{-5}5^2}{2!}$
$\qquad\qquad=\large\frac{0.0067\times 25}{2}$
$\qquad\qquad=0.08375$
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