Probability density functions and Statistical Quantities

When a physical variable, say u can take a wide range of values, it is worthwhile using a notation that distinguishes between the variable itself, u , and its possible values, say v . The probability that the variable u takes a value between v and v + dv is

where f(u) is the probability density function of u . This statement is the definition of PDF, f . The set of all possible values of v is called the sample space. Since the probability that u has a value, any value at all must be 1

where represents integration over the whole of sample space.

Given the probability density function of u , the mean of any function ψ of u is

where the integral is performed over the entire sample space. The averaging is weighted by how often each value v occurs. The mean of variable u itself is

The fluctuating part of u is

Also to be noted

The variance of u is

and the standard deviation of u is the square root of the variance is

which is the rms value of fluctuations. More generally, the nth order central moment of u is

The variance is second-order central moment. The skewness and flatness are normalized higher-order central moments. The skewness of the probability density function u is

and the flatness or kurtosis is

While deriving Reynolds average NS equation or energy equation for turbulent flows, we shall need number of such averaging operations:

where c is a constant

where c is a constant

where c is a constant

or

or

The normal distribution function

The probability density function of the normal distribution is

All higher- order moments of the normal distribution can be expressed in terms of the two lowest-order moment, and

Since the PDF is symmetric, f(-v) = f(v) , so the skewness disappears. The fourth order central moment is

So the flatness is 3.

Task

By making use of the normal distribution as the probability density function, show that the flatness
is 3

We know,

Therefore

put

or or

Now,

or,

Therefore, flatness or Kurtosis =