Question

A radioactive source emits $\mathbf{1800}\mathit{\alpha }$particles during an observation lasting 10 hours. In how many one-minute intervals do you expectno$\mathit{\alpha }\mathbf{\text{'}}\mathit{s}$? $\mathbf{5}\mathit{\alpha }\mathbf{\text{'}}\mathit{s}$?

Short answer

The values required is given below.

$\begin{array}{c}{N}_0=30\\ N_5=60\end{array}$

Step-by-step solution

Given Information  

A radioactive source emits $1800\alpha$ particles during an observation lasting 10 hours.

Definition of the Poisson distribution. 

A discrete frequency distribution that indicates the likelihood of a set of independent occurrences occurring at a specific moment.

Find the values.  

The formula for Poisson distribution states that.$P_n=\frac{{\mu }^n{e}^{-\mu }}{n!}$

The average is given below.

$\begin{array}{l}\mu =\frac{1800}{10\times 60}\\ \mu =3\end{array}$

Probabilities are given below.

$\begin{array}{c}{P}_0=e^{-3}\\ =0.0498\\ P_5=\frac{{\left(3\right)}^5{e}^{-3}}{5!}\\ =0.1\end{array}$

The number of the interval is given below.

$\begin{array}{c}{N}_0=\left(0.0498\right)\left(10\right)\left(60\right)\\ =30\\ N_5=\left(0.1\right)\left(10\right)\left(60\right)\\ =60\end{array}$

The values required is given below.

$\begin{array}{c}{N}_0=30\\ N_5=60\end{array}$