Let the distribution of X be χ2(r).

(a) Find the point at which the pdf of X attains its maximum when r ≥ 2. This is the mode of a χ2(r) distribution.

(b) Find the points of inflection for the pdf of X when r ≥ 4.

(c) Use the results of parts (a) and (b) to sketch the pdf of X when r = 4 and when r = 10.

Answer:

Step 1 of 4:

Let the distribution of ‘X’ be

The probability density function of distribution with ‘r’ degrees of freedom is

Step 2 of 4:

a). Now we have to find the derivative of the pdf of the distribution.

=

=

= ((

f(x) =

Equating the derivative to zero to find the extreme point, that is = 0.

= 0

= 0

Therefore, x = r - 2.

We know that f(0) = 0 , and () can ever be zero at x > 0, so the critical point is x = r - 2. Here the pdf attains is maximum for x = r - 2, where ‘r’ is the degrees of freedom.

Therefore, the mode of distribution is x = r - 2.