Conditional Distribution, for BND. Example 1- Person's Weight and Height

 Conditional Distribution, for BND

Consider you have randomly selected persons from population and measured weight of the person.

Let Y denotes random variable, weight of randomly selected persons in pounds and Y follows normal distribution.

You are interested in determining the probability that a randomly selected individual weighs between 120 and 140 pounds. P(120< Y < 140).

You could imagine that the weight of an individual increase (linearly ?) as height increases.

Thus, you might find it more informative to take into account person's height in calculating the probability that a randomly selected individual weight is between 120 to 140 pounds, for example.

Let X denotes random variable, height of the previous randomly selected persons in inches.

        X follows normal distribution.

Thus, P(120 < Y < 140 | X=x). conditional distribution, is more informative with inclusion of persons height.

To calculate conditional probability, using formula below for BND, you will need to assume/believe

1. Y follows ND
2. The conditional mean of Y given x is linear in x. E(Y|x) is linear in x. Blue line represents the linear relationship between x and condition mean Y.
3. The conditional variance of Y given x is constant. Var(Y|x) is constant, represented by red line.
4. X follows ND.

Based on above, for X, Y condition distribution you can use Bivariate Join Probability Density Function

You can get good three-dimensional graph for above scenario.

Based on theorem, You can understand that, 

If above 4 assumption are meet, then conditional variance can be explained in term of rho as below.

Thus, conditional distribution of Y given X=x is,

Now, with defined condition distribution you can find P(140 < Y < 160 | X=x).

Based on population data, 

X (height in inches) is normally distributed with a mean 67 and variance of 3.76 (Std.dev. 1.939)

Y (weight in pound) is normally distributed with a mean of 127 and variance of 143.06 (Std.dev. 11.961)

Correlation between X and Y is 0.56.

What is the probability that randomly selected person weight is between 120 to 140 pounds given height of 65 inches.

Using BND calculator by SCOR (Statistics Online Computational Resource), School of Nursing, University of Michigan.

P(140 < Y < 160 | X=65) is 0.021.

Visualizing BND, plots below.

Three dimensional plot for conditional probability density functions.

PDF of the Bivariate Normal Distribution, Centered at the origin and Centered elsewhere in the (x, y)-plane.

Reference:

1. 21.1 - Conditional Distribution of Y Given X | STAT 414 (psu.edu)

2. SOCR Data Dinov 020108 HeightsWeights - Socr (ucla.edu)

3. Lesson 4: Multivariate Normal Distribution (psu.edu)

4. Bivariate Normal Distribution -- from Wolfram MathWorld

5. Bivariate Normal Distribution - an overview | ScienceDirect Topics

6. SOCR Bivariate Normal Calculator (umich.edu)

7. SOCR BivariateNormal JS Activity - Socr (ucla.edu)