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# normal distribution

The normal or Gaussian distribution, N(µ,σ), is the distribution of a continuous variable whose probability density function is:

where µ is the mean and σ is the SD and its graph is known as a Gaussian bell curve
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 and SD σ, into a standard normal distribution.
Then, we make the change:

Then the calculus of probability is:
If X = B(n,p) is a binomial variable, then the variable:
approximates to N(0,1) if np≥5 and nq≥5
To calculate the probability of a value we have to make a continuity correction:
Example: In the quality control of a pen drive factory is found that 3% are defective. If we buy 500 pen drives, which is the probability to find 20 or less defective?
Exercises:
1..- The time that an ambulance needs to arrive at a hospital distributes by a normal variable with mean 17 minutes and SD 3 minutes.
a) Calculate the probability that the arriving time is between 13 and 21 minutes
b) Find out the value t, for which the probability that an ambulance takes  more than t minutes to arrive at the hospital is 5%.
2.- A chess player wins 9 out of every 10 games. Calculate the probability that he wins 50 games.
Solutions: 1.- a) 0.8164; b) t≈22minutes; 2.-) 0.122