Joint PDF and CDF of Order Statistics from Standard Normal Distribution

Deriving the Joint PDF and CDF of Ordered Standard Normal Random Variables

Let X1, X2, and X3 be three independent standard normal random variables. We aim to find the joint probability density function (PDF) of Y1, Y2, and Y3, where Y1 < Y2 < Y3 represents the ordered values of X1, X2, and X3. Additionally, we will discuss the approach for calculating the cumulative distribution function (CDF) for each Yi.

Joint PDF Derivation

The joint PDF, fy1Y2Y3(y1, y2, y3), can be expressed as the product of individual conditional PDFs:

fy1Y2Y3(y1, y2, y3) = fY1(y1) * fY2(y2 | Y1=y1) * fY3(y3 | Y1=y1, Y2=y2)

Let's break down each component:

1. fY1(y1): The PDF of the smallest value, Y1. Since any of the Xi could be the smallest, we consider all possibilities:

   fY1(y1) = P(X1=y1, X2>y1, X3>y1) + P(X2=y1, X1>y1, X3>y1) + P(X3=y1, X1>y1, X2>y1) = 3 * fX(y1) * (1 - FX(y1))^2

   Here, fX(y1) is the PDF of a standard normal variable, and FX(y1) is its CDF.

2. fY2(y2 | Y1=y1): The conditional PDF of the middle value, Y2, given Y1. Given Y1, the remaining two variables are distributed like standard normals truncated below at Y1:

   fY2(y2 | Y1=y1) = fX(y2) / (1 - FX(y1)), y2 > y1

3. fY3(y3 | Y1=y1, Y2=y2): The conditional PDF of the largest value, Y3, given Y1 and Y2. This is simply the PDF of a standard normal truncated below at Y2:

   fY3(y3 | Y1=y1, Y2=y2) = fX(y3) / (1 - FX(y2)), y3 > y2

Combining these, the joint PDF becomes:

fy1Y2Y3(y1, y2, y3) = 6 * fX(y1) * fX(y2) * fX(y3), y1 < y2 < y3

CDF Calculation

The CDF, FRi(y), for i = 1, 2, 3, represents the probability that Yi is less than or equal to y. We can obtain these by integrating the joint PDF over the appropriate regions:

• FR1(y) = P(Y1 <= y): Integrate fy1Y2Y3 over the region where y1 <= y, y1 < y2, and y1 < y3.

• FR2(y) = P(Y2 <= y): Integrate fy1Y2Y3 over the region where y2 <= y, y1 < y2, and y2 < y3.

• FR3(y) = P(Y3 <= y): Integrate fy1Y2Y3 over the region where y3 <= y, y1 < y2, and y2 < y3.

Plotting the CDF

Once you've calculated the CDFs for each Yi, you can plot them to visualize the probability distribution of each order statistic.

This analysis provides a comprehensive understanding of the joint distribution and individual probabilities associated with the ordered values of independent standard normal random variables.