Normal Distribution

A Detailed Step-by-Step Explanation

By Bindeshwar Singh Kushwaha
PostNetwork Academy

Introduction: Random Variables

• A random variable (r.v.) is a function that assigns a numerical value to each outcome of a random experiment.
• There are two main types of random variables:
  • Discrete Random Variable: Takes countable values (e.g., number of heads in 3 coin tosses).
  • Continuous Random Variable: Takes any value in an interval of real numbers (e.g., height, weight, time).
• In this section, we focus on the Continuous Random Variable and the most important one: the Normal Distribution.

Definition: Continuous Random Variable

• A random variable \( X \) is said to be continuous if it can take any real value in an interval.
• Its probability is described by a Probability Density Function (PDF) \( f(x) \).
• The probability that \( X \) lies between \( a \) and \( b \) is:
  \( P(a < X < b) = \int_a^b f(x)\,dx \)
• The total area under the curve is always 1:
  \( \int_{-\infty}^{\infty} f(x)\,dx = 1 \)
• The most common continuous distribution is the Normal Distribution.

Definition: Normal Distribution

• • A continuous random variable \( X \) follows a Normal Distribution with mean \( \mu \) and variance \( \sigma^2 \) if its PDF is:

• It is denoted as \( X \sim N(\mu, \sigma^2) \).
• The curve is symmetric about the mean \( \mu \).
• The spread (width) depends on the standard deviation \( \sigma \).

Standard Normal Distribution

• • If \( X \sim N(\mu, \sigma^2) \), we define a new variable:

• Then \( Z \) follows the Standard Normal Distribution:
  \( Z \sim N(0, 1) \)
• Its PDF is given by:
  \( f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2} \)
• This is the most commonly used form for normal probability tables.

Properties of Normal Distribution

• • Mean: \( E[X] = \mu \)
  • Variance: \( \text{Var}(X) = \sigma^2 \)
  • Shape:
    • Bell-shaped and symmetric about \( \mu \)
    • Mean = Median = Mode
    • Area under the curve = 1
  • Empirical Rule:

Graph of Normal Distribution

PDF
NormalDistDef

Applications of Normal Distribution

• Used in natural and social sciences to represent real-valued random variables.
• Commonly applied in:
  • Measurement errors
  • Heights, weights, and blood pressure
  • Quality control and industrial processes
  • Finance (returns, risk modeling)
• The Central Limit Theorem states that the sum of many independent random variables tends to a normal distribution.

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