Mode in a Poisson Distribution with p(x=4) = p(x=2)

The mode of a probability distribution represents the value or values that occur most frequently. In a Poisson distribution, the mode corresponds to the value(s) of X that have the highest probability mass function (PMF) value. To determine the mode, we can compare the PMF values for different values of X in the distribution.

Understanding the Poisson Distribution
The Poisson distribution is a discrete probability distribution that describes the number of events occurring within a fixed interval of time or space. It is often used to model rare events that occur independently at a constant rate. The distribution is defined by a single parameter, λ (lambda), which represents the average rate of event occurrences.

The PMF of the Poisson distribution is given by the formula:

P(X = k) = (e^(-λ) * λ^k) / k!

where k is the number of events, e is the base of the natural logarithm, and k! is the factorial of k.

Comparing p(x=4) and p(x=2)
Given that p(x=4) = p(x=2), we can set up the equations using the PMF formula and solve for the parameter λ.

P(X = 4) = (e^(-λ) * λ^4) / 4!
P(X = 2) = (e^(-λ) * λ^2) / 2!

Since p(x=4) = p(x=2), we can equate the two equations:

(e^(-λ) * λ^4) / 4! = (e^(-λ) * λ^2) / 2!

Simplifying the Equations
To simplify the equations, we can cancel out the common factors:

λ^4 / 4! = λ^2 / 2!

Next, let's simplify the factorials:

4! = 4 * 3 * 2 * 1 = 24
2! = 2 * 1 = 2

Substituting the simplified factorials into the equation:

λ^4 / 24 = λ^2 / 2

Cross-multiplying:

2 * λ^4 = 24 * λ^2

Simplifying further:

λ^4 = 12 * λ^2

Calculating the Mode
To find the mode, we need to solve for the parameter λ that satisfies the equation:

λ^4 = 12 * λ^2

By rearranging the equation, we get:

λ^4 - 12 * λ^2 = 0

Factoring out a common λ^2:

λ^2 * (λ^2 - 12) = 0

Setting each factor equal to zero:

λ^2 = 0 or λ^2 - 12 = 0

The first equation, λ^2 = 0, implies that λ = 0, which is not a valid solution for the Poisson distribution since λ represents the average rate of event occurrences.

Solving the second equation, λ^2 - 12 = 0, we find two possible values for λ:

λ = √12 or λ = -√12

Since the parameter λ cannot be negative, we consider only the positive root

Same doubt, please clear friends and. P4 means??????