Normal Distribution

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A continuous random variable X is normally distributed with parameters and denoted if and only if its density function is: the distribution function is : The Normal distribution depends on two parameters and , which are the expected value and the standard deviation of the random variable X. Expected value, variance and standard deviation: Two important properties of Normal random variables:

  • Linear transformation

    Let X be Normally distributed, and Y be a linear combination of X: . Then, the random variable Y has also Normal distribution:

    Y N(a + b, b )

    The values of the parameters of the transformed random variable follow from the rules for calculating with expecting values and variances:

    E(a + bX) = a + b E(X)

    Var(a + bX) = Var(X) = .

  • Reproduction property

    Let us consider n random variables with Normal distributions:

    The sum of independent, normally distributed random variables , i.e.

    for at least one i, is again normally distributed.

The following diagrams displays a density and distribution function for a N(2;1) random variable. Density:

Nl s2 26 12.gif

The distribution function of N(2;1):

Nl s2 26 13.gif

Standardized random variable: The random variable Z denotes a standardized random variable, which has been centred at its mean and scaled by its standard deviation.If X is normally distributed, then Z also has a Normal distribution. Standardized Normal distribution: The distribution of Z is usually denoted as standardized Normal distribution N(0;1). The density function of a standardized Normal distribution: The distribution function of a standardized Normal distribution: Expected value and variance of standardized Normal distribution: E(Z) = 0 Var(Z) = 1 The density and distribution function for a standardized normal random variable are plotted in the following figures. Density of N(0;1)

Nl s2 26 14.gif

Distribution function of N(0;1)

Nl s2 26 15.gif

The relation between the distribution N() and the standardized Normal distribution: which implies: Confidence interval: A confidence interval for the random variable X is the interval with boundaries and , which will contain the value of the random variable X with probability 1 - , i.e. (1 -) 100% of all values of X will fall in this interval and 100%  will fall outside this interval. 1- is usually referred to as the confidence level.For known values of the expected value of X, the interval is constructed to make the probability that X falls outside this region (there are 2 such regions) with probability /2. We call the interval [] = [] the (symmetric) confidence interval with confidence level P() = 1 - . To stress the importance of the standard deviation, as the parameter of scale, the deviation of X from its expected value is often measured in multiples of . The confidence interval has then this form: [ - c X + c] If the random variable X is N(), then for x = + c the following holds: and P(Z z) = (z) = 1 - /2 . The critical value for the probability 1 - /2 can be obtained from the tabulated values of a standardized Normal distribution. Using these values, we can obtain the confidence interval for a normally distributed random variable: [ X ] and the probability of “this interval”: P( X ) = 1 - The confidence interval for normally distributed random variable:

Nl s2 26 11.gif

We have P(-z Z z) = P(Z z) - P(Z -z) = P(Z z) - [1 - P(Z z)] = 2P(Z z) -1 , which implies that . For given z we can calculate the confidence levels of the interval:

for
for
for

On the other hand, we could also find the value z that produces the desired confidence level 1-, e.g. = 0.95, z = 1.96. The Normal distribution is described by two parameters which imply its:

  • shape
  • location and
  • scale (variance)

In this interactive example, you can choose different values of these parameters and observe their effect on the density function of a normal random variable. We recommend that you only change one parameter at time to better observe their effects on the distribution function. The density function of a standard Normal distribution is presented (in black) to provide a further reference point. In addition, you can also calculate the probability that falls in some interval. Let us consider random variable with Normal distribution .

  1.  We want to compute for :

    Nl s2 26 f 2.gif

    There is a 99.38% probability  that the random variable is smaller than 125.

  2.  We want to calculate the probability for :

    Nl s2 26 f 4.gif

    There is a 5.94% probability that the random variable is greater than 115.6.

  3.  Let us calculate the probability for :

    Nl s2 26 f 6.gif

    The random variable is smaller than 80 with probability of 2.275% .

  4.  Let us compute for :

    Nl s2 26 f 8.gif

    The probability that the random variable is greater than 94.8 is 69.85% .

  5.  We compute the probability for and :

    Nl s2 26 f 10.gif

    The random variable falls in the interval with probability 86.8% .

  6.  Let us calculate for and (centered probability interval):

    Nl s2 26 f 12.gif

    The random variable falls into the interval with probability 95% .

  7.  We want to calculate an interval, which is symmetric around the expected value, such that it will contain 99% of the realizations of :

    For the value (the probability) 0.995 we find in the tables of the distribution function of standard Normal distribution function that .. This implies: take .

    Nl s2 26 f 14.gif

    The random variable falls into the interval with a 99% probability .

  8.  Let us find an such that 76.11% of the realizations of are smaller than :

    For the value 0.7611 we obtain from the standard Normal distribution tables that . Hence: so that .

    Nl s2 26 f 16.gif

    There is a 76.11% probability that the random variable will be smaller than 107.1.

  9.  We calculate such that 3.6% of realizations of is greater than :

    Since , using the standard Normal distribution tables the value for the probability 0.964. Hence, so that .

    Nl s2 26 f 18.gif

    There is a 3.6% probability that the random variable is greater than 118.

The Normal distribution is one of the most important continuous distributions because:

  • approximate normality can be assumed in many applications
  • it can be used to approximate other distributions
  • many variables have normal distributions if there is a large number of observations

A random variable with a Normal distribution can take all values between - and + The Normal distribution is also sometimes referred to as a Gaussian distribution. The density of Normal distribution is sometimes called the Bell curve.The formulas for the density (or the distribution function) imply that a Normal distribution will  depend on the parameters .. By varying these parameters we can obtain a range of distributions. The following diagram shows 5 normal densities with various parameters .

Nl s2 26 m 1.gif

The parameter specifies the location of the distribution. If we change the parameter , the location of the distribution will shift but its shape remains the same.By increasing or decreasing the parameter , the density ”spreads” or ”concentrates”. Large values of , produce flatter and wider densities. Small values of  produce distributions that are narrow and tight. Other properties of the Normal distribution :

  •  the density has global maximum (the mode) at point
  •  the density is symmetric around the point . The symmetry implies that the median is .
  •  the density has inflexion points at and
  •  the density is asymptotically equal to 0 as or .

The digram contains a plot of a distribution

Nl s2 26 m 3.gif

Standard Normal distribution: Tabulating the distribution function of the Normal distribution for all values of and is not possible.However, since we can transform a Normal random variable to obtain another Normal random variable we need only tabulate one distribution. The obvious choice is the Normal distribution with expected value , and standard deviation , . This distribution is called a standard Normal distribution, denoted – distribution. The corresponding random variables are usually denoted by the letter .The random variable is the random variable centered at its mean and divided by its standard deviation. Hence and . If is normally distributed, then also has a (standard) Normal distribution. The standard Normal distribution is important because each random variable with arbitrary Normal distribution can be linearly transformed to a random variable with standard Normal distribution.In most tables for the density and distribution function of the standard Normal distribution, you can find only positive values of . The tables of standard Normal distribution for negative is unnecessary since the Normal distribution is symmetric.