PROBABILITY

# 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 curveEvery 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

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