The number of calls that arrive under a particular time in a cell has been established to be a Poisson distribution. The average number of calls arriving in a cell in 1 millisecond is 5. What is the probability that 8 calls arrive in a cell in a given millisecond?
To find the probability that 8 calls arrive in a cell in a given millisecond, we can use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space, given the average rate of occurrence.
The formula to calculate the probability of observing a specific number of events in a given interval is:
P(x; λ) = (e^-λ * λ^x) / x!
Where:
- P(x; λ) is the probability of observing x events in the interval
- e is the base of the natural logarithm (approximately equal to 2.71828)
- λ is the average number of events in the interval
- x is the number of events
In this case, the average number of calls arriving in a cell in 1 millisecond is given as 5. Therefore, λ = 5.
To calculate the probability of observing 8 calls in a given millisecond, substitute these values into the formula:
P(8; 5) = (e^-5 * 5^8) / 8!
Now we can calculate the probability:
P(8; 5) = (2.71828^-5 * 5^8) / 8!
Using a calculator:
P(8; 5) ≈ 0.1755
Therefore, the probability that 8 calls arrive in a cell in a given millisecond is approximately 0.1755, or 17.55%.
As a side note, the exclamation mark (!) in the formula represents the factorial of a number. The factorial of a non-negative integer x is denoted by x! and is the product of all positive integers less than or equal to x. For example, 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.
The average number of events in a 1 ms interval is a = 5.
Fot m=8 events in that same interval, the probability is
a^m* e^-a/m!
= 5^8*e^-5/8!
=3.906*10^5*6.738*10^-3/4.032*10^4
= 0.065