The Normal or Gaussian Distribution - HAMILTON …

1 The Normal or Gaussian DistributionNovember 3, 2010 The Normal or Gaussian DistributionThe Normal DistributionThe Normal Distribution is one of the most commonly usedprobability Distribution for we repeat an experiment numerous times and averageour results, the random variable representing the average ormean tends to have a Normal Distribution as the number ofexperiments becomes previous fact, which is known as the central limittheorem, is fundamental to many of the statistical techniqueswe will discuss physical characteristics tend to follow a normaldistribution. - Heights, weights, in measurement or production processes can often beapproximated by a Normal variables with a Normal Distribution are said to be normalrandom Normal or Gaussian DistributionThe Normal DistributionThe Normal Distribution is one of the most commonly usedprobability Distribution for we repeat an experiment numerous times and averageour results, the random variable representing the average ormean tends to have a Normal Distribution as the number ofexperiments becomes previous fact, which is known as the central limittheorem.

2 Is fundamental to many of the statistical techniqueswe will discuss physical characteristics tend to follow a normaldistribution. - Heights, weights, in measurement or production processes can often beapproximated by a Normal variables with a Normal Distribution are said to be normalrandom Normal or Gaussian DistributionThe Normal DistributionThe Normal Distribution is one of the most commonly usedprobability Distribution for we repeat an experiment numerous times and averageour results, the random variable representing the average ormean tends to have a Normal Distribution as the number ofexperiments becomes previous fact, which is known as the central limittheorem.

3 Is fundamental to many of the statistical techniqueswe will discuss physical characteristics tend to follow a normaldistribution. - Heights, weights, in measurement or production processes can often beapproximated by a Normal variables with a Normal Distribution are said to be normalrandom Normal or Gaussian DistributionThe Normal DistributionThe Normal distributionN( , ) has two parameters associatedwith it:1 The mean 2 The standard deviation .The probability density functionf(x) ofN( , ) isf(x) =1 2 e (x )22 Normal density function cannot be integrated in closed use tables of cumulative probabilities for a special normaldistribution to calculate Normal Normal or Gaussian DistributionThe Normal DistributionThe Normal distributionN( , ) has two parameters associatedwith it:1 The mean 2 The standard deviation.

4 The probability density functionf(x) ofN( , ) isf(x) =1 2 e (x )22 Normal density function cannot be integrated in closed use tables of cumulative probabilities for a special normaldistribution to calculate Normal Normal or Gaussian DistributionThe Normal Distribution - Properties1 Expected Value:E(X) = for a Normal random :V(X) = : The probability density functionfof a normalrandom variable is symmetric about the mean. Formallyf( x) =f( +x)for all Normal or Gaussian DistributionThe Normal Distribution - Properties1 Expected Value:E(X) = for a Normal random :V(X) = : The probability density functionfof a normalrandom variable is symmetric about the mean.

5 Formallyf( x) =f( +x)for all Normal or Gaussian DistributionThe Normal Distribution 10 8 6 4 (x)The parameter determines the location of the Distribution while determines the width of the bell Normal or Gaussian DistributionThe Standard Normal DistributionThe Normal Distribution with mean 0 and standard deviation 1N(0,1)is called the standard Normal random variable with the standard Normal Distribution is called astandard Normal random variableand is usually denoted cumulative probability Distribution of the standard normaldistributionP(Z z)has been tabulated and is used to calculate probabilities for anynormal random Normal or Gaussian DistributionThe Standard Normal Distribution 3 2 (z)The shape of the standard Normal Distribution is shown Normal or Gaussian DistributionStandard Normal DistributionP(Z z0) gives the area under the curve to the left (z0 Z z1) =P(Z z1) P(Z z0).

6 The Distribution is (Z z0) =P(Z z0).The Normal or Gaussian DistributionThe Standard Normal DistributionExampleSupposeZis a standard Normal random variable. Calculate(i)P(Z );(ii)P(Z> );(iii)P(Z );(iv)P( Z ).(v)P( Z ).The Normal or Gaussian DistributionThe Standard Normal DistributionExample(i)P(Z ): This can be read directly from the (Z ) = (ii)P(Z> ) = 1 P(Z ) = 1 = (iii)P(Z ):Again, we can read this directly from (Z ) = Normal or Gaussian DistributionThe Standard Normal DistributionExample(i)P(Z ): This can be read directly from the (Z ) = (ii)P(Z> ) = 1 P(Z ) = 1 = (iii)P(Z ):Again, we can read this directly from (Z ) = Normal or Gaussian DistributionThe Standard Normal DistributionExample(iv)P( Z ).

7 To calculate this, we note thatP( Z ) =P(Z ) P(Z< )= (v) Similarly,P( Z )=P(Z ) P(Z< )= Normal or Gaussian DistributionThe Standard Normal DistributionExample(iv)P( Z ).To calculate this, we note thatP( Z ) =P(Z ) P(Z< )= (v) Similarly,P( Z )=P(Z ) P(Z< )= Normal or Gaussian DistributionThe Standard Normal DistributionExample(iv)P( Z ).To calculate this, we note thatP( Z ) =P(Z ) P(Z< )= (v) Similarly,P( Z )=P(Z ) P(Z< )= Normal or Gaussian DistributionStandard Normal DistributionExampleDetermine the value ofz0such that:(i)P( z0 Z z0) = ;(ii)P(Z z0) = ;(iii)P( z0 Z z0) =.

8 (iv)P(Z z0) = Normal or Gaussian DistributionStandard Normal DistributionExample(i) IfP( z0 Z z0) = , thenP(Z>z0) +P(Z< z0) = By symmetry, this means thatP(Z>z0) = orP(Z z0) = the table of cumulative Normal probabilities, the value ofz0is (ii) This time, we require thatP(Z z0) = the table again, we find that the value ofz0is Normal or Gaussian DistributionStandard Normal DistributionExample(i) IfP( z0 Z z0) = , thenP(Z>z0) +P(Z< z0) = By symmetry, this means thatP(Z>z0) = orP(Z z0) = the table of cumulative Normal probabilities, the value ofz0is (ii) This time, we require thatP(Z z0) = the table again, we find that the value ofz0is Normal or Gaussian DistributionStandard Normal DistributionExample(iii) As in part (i), we are looking for a valuez0such thatP(Z z0) = the table of Normal probabilities, the value ofz0is (iv) Finally, using the table, the value ofz0for whichP(Z z0) = Normal or Gaussian DistributionStandard Normal DistributionExample(iii) As in part (i), we are looking for a valuez0such thatP(Z z0) = the table of Normal probabilities, the value ofz0is (iv) Finally, using the table, the value ofz0for whichP(Z z0) = Normal or Gaussian DistributionStandard Normal DistributionExample(iii) As in part (i)

9 , we are looking for a valuez0such thatP(Z z0) = the table of Normal probabilities, the value ofz0is (iv) Finally, using the table, the value ofz0for whichP(Z z0) = Normal or Gaussian DistributionStandardisingThe key fact needed to calculate probabilities for a general normalrandom variable is the X is a Normal random variable with mean and standarddeviation , thenZ=X is a standard Normal random means that to calculateP(X x) is the same as calculatingP(Z x ).The Normal or Gaussian DistributionNormal Distribution ExamplesExampleThe actual volume of soup in 500ml jars follows a normaldistribution with mean 500ml and variance 16ml. IfXdenotes theactual volume of soup in a jar, what is(i)P(X>496)?

10 ;(ii)P(X<498)?;(iii)P(492<X<512)?(iv)P(X>480)?The Normal or Gaussian DistributionNormal Distribution ExamplesExample(i)P(X>496) =P(Z>496 5004)=P(Z> 1) = 1 = (ii)P(X<498)=P(Z<498 5004)=P(Z< ) = Normal or Gaussian DistributionNormal Distribution ExamplesExample(i)P(X>496) =P(Z>496 5004)=P(Z> 1) = 1 = (ii)P(X<498)=P(Z<498 5004)=P(Z< ) = Normal or Gaussian DistributionNormal Distribution ExamplesExample(iii)P(492<X<506) =P(492 5004<Z<506 5004)=P( 2<Z< )=P(Z< ) P(Z 2)= = (iv)P(X>493) =P(Z>493 5004)=P(Z> )= 1 P(Z ) = 1 = Normal or Gaussian DistributionNormal Distribution ExamplesExample(iii)P(492<X<506) =P(492 5004<Z<506 5004)=P( 2<Z< )=P(Z< ) P(Z 2)