Normal Distribution

Continuous Random Variables

Definition

A continuous random variable is a random variable whose value can take any number (real number) in an interval

Example

   1 The weight of a randomly chosen student in a class is a continuous random variable.

   2 The time taken by a given student to get to school on a randomly chosen day is a continuous random variable.

   3 The future price of a given stock (assuming it does not default/go bankrupt) is a continuous random variable.

The probability distribution of a continuous random variable X , is uniquely described by its cumulative distribution function (CDF), and is denoted by FX . This is the same as discrete random variables.

F_X (x ) = P(X ≤ x )

Since X is continuous, the probability of X being equal to any point is zero, that is P(X = x ) = 0 for any real number x . This means that when denoting the probability involving the values of a continuous random variable, non-strict inequalties are equivalent to the respective strict inequalities. For example,

P(X ≤ x ) = P(X < x )                and       P(X ≥ x ) = P(X > x )

The continuity of the random variable also implies that the probability mass function of X is zero anywhere, and therefore is not useful to describe the probability distribution of X .

 

Probability Density Function

Definition

The probability density function (PDF) of a continuous random variable is defined as the density of the probability distribution of the random variable. The probability density function of a continuous random variable X is defined as

d

f_X (x ) = dt F_X (t)|_t=x

where F_X is the cumulative distribution function.

 

The probability density function can be seen as an equivalent to the probability mass function of a discrete random variable. However, it is wrong to write the equation.

f_X (x ) = P(X = x )

This is because P(X = x ) is equal to 0 for any real number x , but the left hand side of the equation is not.

The normal distribution or Gaussian distribution is a continuous probability distribution and is the most commonly used in statistics. Normal distributions are important and are often used in the natural and social sciences to represent real valued random variables whose distributions are not known. A random variable with a normal distribution is said to be normally distributed and is called a normal deviate.

The normal distribution is sometimes informally called the bell curve due to the shape of its distribution curve. However, some other distributions are also bell shaped.

When a random variable X follows a normal distribution with parameter, µ and σ2, it is written as X ∼ N(µ, σ2). The parameters µ and σ2 are the mean and variance of the random variable respectively.

The normal distribution has the following properties

1.       The distribution is symmetrical about the mean.

2.       The mean, median and mode of the distribution are equal.

3.       The probability density tails off fairly rapidly as the values of the variable move further away from the mean.

 

The following figure shows the probability distribution for different values of µ and σ².

The distribution with greater mean is centred on the right of the one with less mean. The distribution with greater variance has spread greater than one with less variance, and therefore its graph has flatter curve.

The following figure shows the probability distribution of a normal random variable.

For the normal distribution, the values less than one standard deviation away from the mean account for approximately 2 of the set, while two standard deviations from the mean account for approximately 95%, and nearly all the values lie within three standard deviations away from the mean.