1. Define Normal distribution. What are the main characteristics of a Normal distribution?
  2. What do you mean by probability density function? Write down its properties.

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Answer a):

Binomial distribution and possion distribution are discrete probability distributions. Since, the variables under study were discrete random variables. Normal distribution also known as Normal probability distribution or Gausion distribution is based on the continuous random variable which is used to describe the theory of accidental errors of measurements. Today, it is one of the most important probability models in statistical analysis.

Characteristics:

  1. It is a continuous probability distribution within parameter μ and σ2
  2. Normal curve is bell shaped curve and symmetrical about mean (μ)Normal Distribution | HAMROCSIT
  3. The distribution of curve is max when x = μi.e. f(x) = \(\frac{1}{σ \enspace . \enspace \sqrt{2x}}\)
  4. Since, the curve us symmetrical. So, mean = median = mode
  5. The cofficient of skewness for normal curve is zero. i.e. β1 = 0 and Y1 = 0and the cofficient of kurtosis, β2 = 3 and Y2 = 0

    Here, \(z = \frac{x – μ}{σ}\) is called standard normal distribution. E(z) = 0, var(z) = 1

Answer b):

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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