## Tuesday, February 12, 2013

### Properties of Normal Distribution

Definition:

A set of variable data the theoretical frequency is represented by bell shaped curve which is symmetrical about the mean. We can also call it as Gaussian distribution. The Normal distribution has the peak at the mean. The properties of normal distribution are related to normal distribution curve. The normal distribution is a continuous probability description and it gives the good description of data that cluster around the mean. Properties of normal distribution function is used to understand the normal distribution.

Properties of normal distribution

The normal curve is given by the following formula:

f(x) =   e- (X-?)2/(2?2)  / (? `sqrt(2Pi)`)

Properties of normal distribution are having eight types. Here the list of Properties is available.

Property 1:

The normal distribution is closed under the linear transformation. Where X is a  normally distributed random variable with mean  and standard deviation . Then the normal distribution aX+b is also distributed (where for some real numbers a???0 and b)

aX + b ~`aleph`( a ? + b , a2 ? 2 )

And X1, X2 are the two independent normal random variables and which is having the mean ?1, ?2 and standard deviations ?1, ?2       then their linear combination also distributed.

Property 2:

If  X1, X2     are the two independent random variable then their Sum X1+ X2     are also distributed

Property 3:

The above property is sometimes mistakenly believed that two random variables are uncorrelated and independent. But it is not true. The correct property is the two random variables are “Jointly Normal”  and uncorrelated and also independent. Is this topic Calculate Harmonic Mean hard for you? Watch out for my coming posts.

Property 4:

Normal distribution is infinitely divisible. For a normally distributed random variable X with mean and standard deviation  we can find n independent random variables {X1, …, Xn} each distributed normally with means ?/n and variances ?2/n  such that

X1+X2+..............................+Xn ~ `aleph` (?, ? 2)

Property 5:

Normal distribution is stable. If X1, X2 are two independent random variable  N(?,??2 ) and a, b are arbitrary real numbers, then

aX1+bX2 ~ `sqrt(a^2 + b^2)` .X3 + (a + b - `sqrt(a^2 + b^2)` )?

Property 6:

The normal distribution belongs to the exponential family with the parameters ?1 =? /   ? 2 and ?2 = -1/   ? 2

Property 7:

The normal distribution family forms a manifold with a constant curvature -1.Tha same family is flat respective to ±1 for the connections ?(e) and ?(m)

Property 8:

For the fisher man matrix the normal distribution is diagonal and it takes the form of

`[[1 / sigma^2,0],[0,1 / (2sigma^4)]]`

The above properties of normal distribution is used to describe the normal distribution curve.