- What do you understand by Poisson distribution? What are its main features?
- What do you mean by joint probability distribution function? Write down its properties.

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**a) Possion Distribution:**

In Statistics, Poisson distribution is one of the important topics. It is used for calculating the possibilities for an event with the average rate of value.

The Poisson distribution is a discrete probability function that means the variable can only take specific values in a given list of numbers, probably infinite. A Poisson distribution measures how many times an event is likely to occur within “x” period of time.

The discrete random variable X is said to follow Poisson distribution if its probability mass function is given by

P(X=x) = P(x) = \(\frac{e^{-λ} λ^{x}}{x!}\), x=0,1,2,3,……….

Here, λ is the parameter of poisson distribution and a random variable dollowing poisson distribution is denoted by X ~ P(λ).

Some examples of Poisson Distribution are:

- Number of death from a disease
- Number of suicides in particular interval of time in particular locality.
- Number of misprints in a page of a book
- Number of telephone calls in given interval of time

Properties of Poisson Distribution:

- Poisson distribution is discrete distribution.
- It has only one parameter λ, hence it is also known as uni-parametric.
- Mean = λ
- Variance = λ
- Mean = Variance
- For non-integer λ, it is the largest integer less than λ. Fir integer λ, x = λ and x = λ – 1 are the two modes.
- Coefficient of Skewness β
_{1}=1/λ, γ_{1}=1/√λ - Coefficient of Skewness β
_{2}=1/λ, γ_{2}=1/√λ - Sum of two Poisson variate is Poisson variate i.e. if X ~ P(λ
_{1}), X_{2}~ P(λ_{2}) then X_{1}± X_{2}~ P(λ_{1}+ λ_{2}). - IT is used in the case of waiting time analysis where np < 5 and probability of success is very low i.e. p → 0 and n → ∞.

**b) Probability Density Function:**

Let x be a continuous random variable and f(x) is the continuous probability function of x and it is also known as probability density function.

Properties:

Let x be the continuous random variable with density function f(x), and the probability density function should satisfy the following conditions:

- For a continuous random variable that takes some value between certain limits, say a and b, the PDF is calculated by finding the area under its curve and the X-axis within the lower limit (a) and upper limit (b). Thus, the PDF is given by\(P(x) = \int_a^b f(x) dx\)
- The probability density function is non-negative for all the possible values, i.e.
**f(x)≥ 0**, for all x. - The area between the density curve and horizontal X-axis is equal to 1, i.e.\(\int_{-∞}^{∞} f(x) dx = 1\)
- Due to the property of continuous random variables, the density function curve is
**continued**for all over the given range. Also, this defines itself over a range of continuous values or the domain of the variable.

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