Let X be a Poisson random variable with parameter λ=5 Compute the probabilities p0, …, p6 to fou

step 1 Welcome to numerate. In the current problem, lambda is equal to 5 and we have to find the probab

Let X be a Poisson random variable with parameter λ=5...

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space when these events happen with a known constant rate and independently of the time since the last event. It is widely used in fields such as statistics, physics, and quality control for modeling count data that occur randomly over a continuous parameter.

Probability Mass Function (PMF)

The PMF for the Poisson distribution gives the probability of observing exactly k events, and it is expressed as P(X = k) = (e^(-?) * ?^k) / k!. This function involves the parameter ?, which signifies the average rate of occurrences, and incorporates factorial calculations and the exponential function, which are critical for computing the probabilities accurately.

Parameter (?)

The parameter ? represents the mean or expected number of occurrences within the fixed interval and plays a central role in the Poisson distribution by scaling the probability mass function. It serves as a measure of the intensity or rate at which the events occur, and variations in ? directly affect the spread and the shape of the distribution.

Computation with Rounding

When calculating probabilities for a Poisson random variable, it is common to perform numerical computations using the PMF and then round the results to a specified number of decimal places. This practice is important in practical applications to ensure that the probabilities are presented in a clear and interpretable format, suitable for reporting and further analysis.

Transcript

00:01 Welcome to numerate.

00:01 In the current problem, lambda is equal to 5 and we have to find the probabilities for the values of x equals to 0 1 to 2 .0 .0 till 6.

00:14 Okay, so we have to find values p0, p1, everything.

00:18 Now, we know the poizzo pmf is e to the power minus lambda into lambda to the power x divided by x factorial, which is e to the power.

00:30 Power minus 5 into 5 to the power x by 5 factorial.

00:36 Now, if you want to find for 0, you have to put 0 over here and, oops, not 5 factorial, it is x factorial, right, and n over here, two places.

00:50 And for that, you have to write each and every term on the calculator.

00:54 So rather than going that way, we will take another approach.

00:57 But before that, we will just find out p0.

01:00 That would be to the power minus 5 into 5 to the power 0 by 0 factorial.

01:07 Now this and this both are 1 and e to the power minus 5 would be 0 .00678.

01:18 Okay.

01:20 Now following the same procedure we can find 1, p2, p3 but i want to be little faster in the examination hall.

01:28 So i don't want to do all those calculations.

01:31 What i will do is i would rather have this recursion relation remembered.

01:39 Okay, because if you see what will be p1, p1 will be, okay, instead of lambda, okay, no, this is lambda only...

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