Joint and Range Distribution from a Continuous Population

Joint Probability Distribution

Property 1: Let F1,n(x,y) be the joint distribution function for the first and last order statistic for a sample of size n taken from a population with cdf F(x). Then for x < y

Joint distribution function

Proof:

Proof 1

By Property 1 of Distribution of Order Statistics from a Continuous Distribution, the cdf of the nth order statistic is

Proof 2We now have

F(x,y)

Last line of proof

Property 2: Let f1,n(x,y) be the joint pdf function for the first and last order statistic for a sample of size n taken from a population with cdf F(x) and pdf f(x). Then

f_1,n(x,y)

if x < y. Otherwise, f1,n(x,y) = 0.

Property 3: Let fj,k(x,y) be the joint pdf function for the jth and kth order statistic for a sample of size n taken from a population with cdf F(x) and pdf f(x). Then

f_j,k(x,y)

if x < y and j < k. Otherwise, fj,k(x,y) = 0.

Observation: When j = 1 and k = n, by Property 3

f_1,n(x,y) part 1

f_1,n(x,y)

which is the same result we obtained in Property 2.

Property 4: Let Fj,k(x,y) be the joint distribution function for the jth and kth order statistic, with j < k, for a sample of size n taken from a population with cdf F(x) and pdf f(x). Then for x < y

F_j,k(x,y)

and for y ≤ x

F_j.k(x,y)

Proof:

Proof 1

Proof 2

Proof 3

Proof 4

Range Distribution

The range of a sample of size n is x(n) – x(1). This definition can be extended to x(k) – x(j).

Property 5: The pdf h(w) of w = x(k) – x(j) is equal to the area under the curve

y = fj,k(x,x+w)

Observation: Based on Properties 3 and 5, the pdf of w = x(n) – x(1) is equal to the area under the curve

y = n(n-1)[F(w+x)–F(x)]n-2f(w+x)f(x)

where F(x) is the cdf of the population and f(x) is the pdf of the population.

Property 6: The cdf of w = x(n) – x(1) is equal to the area under the curve

y = n[F(w+x)–F(x)]n-1f(x)

Proof: Click here for the proof that uses calculus.

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack supports the following worksheet functions. These functions refer to a distribution dist (“uniform”, “normal”, etc.) with the specified parameters as described for the MEAN_DIST and VAR_DIST functions (see Distribution Property Functions).

ORDER2_DIST(x, y, j, k, n, cum, dist, param1, param2, param3) = the pdf at (x, y) for the jth and kth order statistic from a sample of size n for the specified distribution if cum = FALSE and the corresponding cdf F(x) if cum = TRUE.

RANGE_DIST(x, j, k, n, cum, dist, param1, param2, param3) = the pdf at x for the range x(k) – x(j)  from a sample of size n for the specified distribution if cum = FALSE and the corresponding cdf if cum = TRUE.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Chen, P-N (2008) Basic theories on order statistics
Reference is no longer available

Ma D. (2010) The distribution of the order statistics. A Blog on probability and statistics
https://probabilityandstats.wordpress.com/2010/02/20/the-distributions-of-the-order-statistics/

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