Download Normal, Student`s, F and Chi

Important Distribution Functions
Dr. Dennis S. Mapa
Prof. Manuel Leonard F. Albis
UP School of Statistics
Normal Distribution
 The Normal Distribution is the most important
continuous distribution.
 This distribution represents adequately many random
processes.
 This has a bell-like shape with more weight in the
center and tails tapering off to zero.
 The normal distribution can be characterized by its
first two moments, the mean µ (location) and the
variance 2 (dispersion).
Normal Distribution
A continuous random variable X is said to be normally
distributed if its density function is given by ,
f ( x) 
1
2 2
 1  x   2 
exp 
 
 2    
for - < x < 
Its mean is E[X] = µ and variance is V[X] = 2.
We denote the distribution as N(µ,2).
The Normal Distribution
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
u-3s u-2s
u-s
u
u+s u+2s u+3s
 distribution of values is
bell-shaped
 symmetric about the
population mean values
(µ)
 both tails of the curve
will
approach
the
horizontal axis but will
never touch it
The Normal Distribution
In any normal distribution, observations are distributed
symmetrically around the mean, with,
 about 68.27% of all values under the curve lie
within one standard deviation of the mean (µ ± )
 about 95.45% of all values under the curve lie
within two standard deviation of the mean (µ±2)
 about 99.73% of all values under the curve lie
within three standard deviation of the mean
(µ±3)
The Normal Distribution with
Different Means and Variances
The Standard Normal Distribution
Instead of having to deal with different parameters, it
is often convenient to work with the standard normal
variable, Z.
The standard normal distribution has a mean 0 and
variance of 1 (standard deviation is also 1).
If X is normally distributed with mean µ and standard
deviation , X may be transformed into Z = (X - µ)/,
where Z is standard normal.
Central Limit Theorem
If X is the mean of a random sample of size n taken
from a large (or infinite) population with mean  and
variance 2, then the sampling distribution of X is
approximately normally distributed with mean  and
variance 2/n when n is sufficiently large.
Hence, the limiting form of the distribution of
X 
Z
 n
is the standard normal distribution.
The Student’s t Distribution
 The Student’s t distribution is an important
distribution that we use in hypothesis testing.
 It describes the distribution of the ratio of the
estimated coefficient and its standard error.
 The distribution is characterized by the parameter k
known as the degrees of freedom. Its probability
density function is given by,
k  1 / 2 1
1
f ( x) 
k / 2
k 1  x 2 / k


( k 1) / 2
The Student’s t Distribution
 The distribution is symmetrical with mean zero and
variance V(X) = k/(k – 2), provided k > 2.
 Its kurtosis is  = 3 + 6/(k – 4), provided k > 4.
 It has fatter tails than the normal distribution which
often provides a better representation of typical
financial variables.
 k is usually between 4 to 6.
The Student’s t Distribution
Student’s t – distribution
 If X and S2 are the sample mean and sample variance,
respectively, of a random sample of size n taken from a
population which is normally distributed with mean  and
variance 2, then
X 
T
S n
is a random variable having the t - distribution with v = n-1
degrees of freedom.
The t-distribution is an important distribution in hypothesis
testing.
F- distribution
 The F distribution is an asymmetric distribution that has a minimum value of
0, but no maximum value.
 The curve reaches a peak not far to the right of 0, and then gradually
approaches the horizontal axis as the F value becomes large. The F
distribution approaches, but never quite touches the horizontal axis.
 The F distribution has two degrees of freedom, d1 for the numerator, d2 for
the denominator. For each combination of these degrees of freedom there is a
different F distribution.
 The F distribution is most spread out when the degrees of freedom are small.
As the degrees of freedom increase, the F distribution is less dispersed.
F- distribution
 The figure below shows the shape of the distribution. The F
value is on the horizontal axis, with the probability for each F
value being represented by the vertical axis. The shaded area in
the diagram represents the level of significance α shown in the
table.
Chi Square Distribution
 In probability theory and statistics, the chi-square
distribution (also chi-squared or χ²-distribution) with k
degrees of freedom is the distribution of a sum of the squares
of k independent standard normal random variables.

 The chi-square distribution is a mathematical distribution that
is used directly or indirectly in many tests of significance.
 The chi-square distribution has one parameter, its degrees of
freedom (df). It has a positive skewness.
 As the df increase, the chi square distribution approaches a
normal distribution.
Chi Square Distribution
Examples of Chi-Square Distribution with varying degrees of
freedom