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According to the Poisson distribution formula, we find out that the probability of less than 3 occurrences for the given random variable is 0.1016.
It is given to us that -
The random variable x is the number of occurrences of an event over an interval of ten minutes
It can be assumed that the probability of an occurrence is the same in any two time periods of an equal length
The mean number of occurrences in ten minutes is 5.3
We have to find out the probability that there are less than 3 occurrences in ten minutes.
Firstly, we can determine that the given random variable x follows Poisson distribution.
This is because Poisson distribution is referred to as a discrete probability distribution which helps us to determine the probability of a certain number of events that occur within a specific time interval.
Here, we see that the random variable x provides the number of occurrences of an event with a specific time interval of ten minutes. So, x follows Poisson distribution.
We know the formula for Poisson distribution is given as -
p(x) = {[e^(-λ)] * (λ^x)}/x! ---- (1)
where,
λ = mean number of occurrences within given time interval
x = number of occurrences
From the given information, we can say that -
λ = 5.3 and x = 3
Substituting these values of λ and x in equation (1), we have
p(x) = {[e^(-λ)] * (λ^x)}/x!
[tex]= > p(x)=\frac{(e^{-5.3} )*(5.3^{3} )}{3!} \\= > p(x) = 0.1016[/tex]
Thus, by applying Poisson distribution formula, we find out that the probability of less than 3 occurrences for the given random variable is 0.1016.
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