Statistics for decision

This is a very common law, frequently observed in nature. E.g. the height of the human population follows a normal distribution: many individuals have a height close to the mean height of the population and few individuals are either very short or very tall.

A variable U follows a standard normal distribution if its density on real line is defined as:

If μ is a real number and σ a positive real number, the Gaussian distribution, or normal distribution N (μ, σ) of the density on is defined as:

In such case, we note that μ is the mean (also called expectation), and σ the standard deviation.

As far as calculation is concerned, we resort to tables.

Let us have a random variable X, following a normal distribution N (μ ,σ). We may find an interest in calculating the probability for X to have a value belonging to the interval [a,b] (it could be the income range of the people you target as a sales person, or the range of shoe sizes you plan to keep on stock).

Thus, .

In order to calculate this integral, we introduce the variable .

The integral becomes

with .

We realise the density obtained after the change of variable is that of the standard normal distribution.

Therefore, if we define U as a random variable such that , U follows a standard normal distribution N (0 ,1) and

.

But these integral calculations are not always straightforward ( we strongly advise against trying to find a primitive to the Gaussian density). To solve such calculation, we use standard normal distribution tables displaying results for pre-set values.

We now understand the necessity to standardise the distribution (i.e. to consider the U variable rather than the X variable), since it is impossible to pre-calculate tables for every possible parameter (μ ,σ).

E.g. tables provide values for the cumulative distribution function of the random variable U following a standard normal distribution (@@@INSERER référence à la table ), meaning the function :

.

This function has the following properties :

(−u) = 1 −(u)

(0]=0,5

as a consequence .

Refer to the table link here to find out how to read it and make calculations.

Another interesting activity is to express the deviation of the X variable from the mean as a multiple of the standard deviation σ.

Let us try and compute, for k>0 :

.

It is important to remember a remarkable value in the tables: P (−1, 96 < U < 1, 96) = 0, 95. The value 1.96 is important because if we decide to say: “The variable U belongs to the interval [-1.96; 1.96]”, we know we take a risk of 5% to be wrong.

Similarly, with a risk of 1% of error, U belongs to the interval [-2.576; 2.576].

There are also tables to find these 1.96 and 2.576 figures, they are called fractiles tables.

We use the notation for the fractile of order , i.e. the number such that

.

The fractiles provide invaluable information.

Let us imagine the range X of a TRF1 artillery canon follows a normal distribution with the parameters μ =24km and σ=25m

Proposition 1.10 let X be a random variable following a normal distribution N (μ ,σ), we have :

P (μ − 1, 96σ < X < μ+ 1, 96σ) = 0, 95.

We can then say that 95% of the shells will fall in a zone between 24km-50m and 24km+50m.

If we wish to find the zone where 99% of the shells will fall, we use the fractile of order 0.995, the table gives a value of 2.57.

We get P (μ − 2, 57σ < X < μ+ 2, 57σ) = 0, 99 and conclude that 99% of the shells will fall in a zone between 24km-65m and 24km+65m.

Tir TRF1

This demonstrate the huge benefits we get if we can demonstrate that a variable follows a normal distribution. For this there are normality tests, that allow us to use or reject the hypothesis that a random variable follows a normal distribution.

Let us consider ν independent, random variables U₁, U₂, · · ·, U_v, each following a standard normal distribution N (0 ,1). We will name Chi-squared distribution with ν degrees of freedom the distribution followed by the variable . We shall use the notation for this distribution law.

Its density function is defined as:

.

Its mean (or mathematical expectation) is ν, and its variance is 2 ν.

As for the normal distribution, we shall use the tables linked here (@@@@INSERER REFERENCE).

Densités des lois du Khi-deux ddl=2,10,20,40

Let U and X be two independent variables. U is following a standard normal distribution N (0 ,1) and X is following a Chi-squared distribution with ν degrees of freedom.

The variable T defined as is following a Student distribution with ν degrees of freedom.

Its mean (or mathematical expectation) is 0, and its variance is .

One can notice that the Student distribution offers an uncanny resemblance to the standard normal distribution when ν is large. It is indeed possible to assimilate the student distribution to the standard normal distribution.

Again, refer to the tables linked here (@@@@INSERER REFERENCE).

Densités des lois de Student ddl=1,3,50

Let X₁and X₂ be two independent variables, each following a Chi-squared distribution with respectively ν₁ and ν₂ degrees of freedom. Then the F variable, defined as

follows a Fischer-Snedecor distribution with ν₁ and ν₂ degrees of freedom, denoted F(ν₁, ν₂).

Its mean (or mathematical expectation) is and its variance is for ν₂ with large enough values.

Fischer-Snedecor distribution tables are available here. (@@@@@INSERER REFERENCE)