1. What do you understand by Poisson distribution? What are its main features?
  2. 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:

  1. Number of death from a disease
  2. Number of suicides in particular interval of time in particular locality.
  3. Number of misprints in a page of a book
  4. Number of telephone calls in given interval of time

Properties of Poisson Distribution:

  1. Poisson distribution is discrete distribution.
  2. It has only one parameter λ, hence it is also known as uni-parametric.
  3. Mean = λ
  4. Variance = λ
  5. Mean = Variance
  6. 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.
  7. Coefficient of Skewness β1=1/λ,  γ1=1/√λ
  8. Coefficient of Skewness β2=1/λ,  γ2=1/√λ
  9. Sum of two Poisson variate is Poisson variate i.e. if X ~ P(λ1), X2 ~ P(λ2) then X1 ± X2 ~ P(λ1 +  λ2).
  10. 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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