Sampling

# 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.

We use this case to calculate other possibilities:

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

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