Normal distribution

As you know, every distribution with mean μ and standard deviation σ can be associated to a normal or gaussian distribution with mean 0 and standard deviation 1.

This is the standard normal distribution or unit normal distribution N(0,1). The random variable associated to this distribution, Z, is called standard normal deviate.

We use a table with the values of P(Z ≤ a), a > 0, which is the area of the shaded zone.

For using the table, we have to convert the variable X which follows a distribution with mean μ into a standard normal distribution.

Then, we make the change:

Then the calculus of probability is:

The binomial distribution, B(n,p), has this function of probability:

 

With mean and standard deviation:

 

If X = B(n,p) is a binomial variable, then the variable:

   approximates to N(0,1)

 

 

 

Exercise: If X = N(20,2), calculate:

 

a) P (X > 24)

 

b) P (21 < X < 25)

 

 

 

 

 

 

 

Solution: a) 0.0228; b) 0.3023