# Conditioning of join gaussian over a line

I need to compute the conditional probability of bivariate normal distribution over a line. Let's suppose that X and Y both are normal distributions and that they are independent. Let's suppose that we want to find the probability that a point 'k' will be on the line L= aX+bY=c. To do so we need to compute the following:
$$p(k|aX+bY=c) = \frac {p_x(X_k)p_y(Y_k)}{p(aX+bY=c)} = \frac {p_x(X_k)p_y(Y_k)}{\int_{L} p_x(X)p_y(Y) dX dY}$$

where k is point over the line L. For this to happen we will use a base for the line L define as a vector b(b1,b2) and a point of origin r(r1,r2). Therefore, the previous integral can be written as:
$$\frac {p_x(X_k)p_y(Y_k)}{\int_{t} p_x(r_1+b_1t)p_y(r_2+b_2t) dt} = \frac {p_x(r_1+b_1k)p_y(r_2+b_2k)}{\int_{t} p_x(r_1+b_1t)p_y(r_2+b_2t) dt}$$

My goal now is to show that this last fraction is indeed a gaussian distribution and to get its mean and variance

• (1) It is essential that $(X,Y)$ have a bivariate Normal distribution: marginal (univariate) Normal distributions do not give sufficient information. (2) Assuming that, and assuming it is nondegenerate, then there exist $a^\prime$ and $b^\prime$ (easily found in terms of the covariance matrix of $(X,Y)$) for which $(U,V)=(aX+bY, a^\prime X+b^\prime Y)$ are independent and Binormal, reducing the question to a triviality.
– whuber
May 12 at 21:35
• @whuber I tried to make my question clearer. Could explain what said differently?
– sam
May 12 at 21:39

A bivariate normal density can be likened to a piece of bologna (or did I mean to write baloney?) about which Americans often say "No matter how you slice it, it is still bologna". The meaning w.r.t. bivariate normal densities (regarded as a piece of bologna sitting on the $$x$$-$$y$$ plane) is that every cross-section of this piece (of lunch meat) has the shape of a univariate normal density. Slice it along the straight line $$ax+by=c$$ and the cross-section has the shape of a normal density. It is not exactly a univariate normal density in that the "area under the curve" in usually not $$1$$ as it must be for all valid univariate densities, but for any nonnegative function $$g(x)$$ with finite area $$A$$, $$A^{-1}g(x)$$ is a valid density function. So, as I said, the shape is of the correct form, but the area need not be, but the area is readily fixed.

So, what is the density of $$aX+bY+c$$ when $$(X,Y)$$ is bivariate normal? Easy-peasy -- it is a normal density with mean $$E[aX+bY+c] = a\mu_x +b\mu_Y +c$$ and variance $$\operatorname{var}(aX+bY+c) = a^2\operatorname{var}(X) + b^2\operatorname{var}(Y) +2ab\operatorname{cov}(X,Y).$$

• thank you for your answer. So is the result oof this :$\frac {p_x(r_1+b_1k)p_y(r_2+b_2k)}{\int_{t} p_x(r_1+b_1t)p_y(r_2+b_2t) dt}$ is a gaussian with the specified parameters? and how those parameters relate to the base that I defined for the line?
– sam
May 13 at 5:13
• This is an excellent analysis. However, the question doesn't ask for this marginal distribution: it requests the conditional distribution.
– whuber
May 13 at 12:10

Let $$X$$ and $$Y$$ be jointly normal random variables with means $$\mu_X, \mu_Y$$, and covariance matrix $$\Sigma$$. (We do not need that $$X$$ and $$Y$$ are independent, although it does simplify some calculations.)

$$\begin{pmatrix}X\\Y \end{pmatrix} \sim N(\begin{pmatrix}\mu_X \\ \mu_Y \end{pmatrix} , \begin{pmatrix}\Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22}\end{pmatrix})$$

Pre-multiply by a matrix $$A$$ to transform the second variable into the 'variables-part' of desired line equation, $$aX + bY$$. As $$A$$ is fixed and $$X,Y$$ are jointly normal, the result is still jointly normal.

$$A\begin{pmatrix}X\\Y \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ a & b \end{pmatrix}\begin{pmatrix}X\\Y \end{pmatrix} = \begin{pmatrix}X\\ aX + bY \end{pmatrix} \sim N(A\begin{pmatrix}\mu_X \\ \mu_Y \end{pmatrix} , A\begin{pmatrix}\Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22}\end{pmatrix}A^T)$$

The mean of this multivariate distribution is

$$\begin{pmatrix} \mu_X \\ a\mu_X + b\mu_Y \end{pmatrix}$$

and, for simplicity, denote the resulting variance as

$$A\begin{pmatrix}\Sigma_{11} & \Sigma_{12}\\ \Sigma_{21} & \Sigma_{22}\end{pmatrix}A^T = \begin{pmatrix}\Omega_{11} & \Omega_{12}\\ \Omega_{21} & \Omega_{22}\end{pmatrix}$$

Finally, follow one of the answers in this question, Deriving the conditional distributions of a multivariate normal distribution, to prove that the distribution of $$X$$ conditional on $$aX + bY = c$$ has a normal distribution with mean

$$\mu = \mu_X + \Omega_{12}\Omega_{22}^{-1}(c - a\mu_X - b\mu_Y)$$

and variance

$$\sigma^2 = \Omega_{11} - \Omega_{12}\Omega_{22}^{-1}\Omega_{21}$$

We now have the conditional distribution of $$X$$ given $$aX + bY = c$$. The corresponding value of $$Y$$ can be easily found from the same line equation, $$Y = c/b - (a/b)X$$.

For a point $$k$$ along the line $$(r_1 + b_1k, r_2 + b_2k)$$ The conditional distribution of $$X$$ can be re-expressed to give the conditional distribution of $$K$$, by re-arranging $$x = r_1 + b_1k$$ to $$k = (x - r_1)/b_1$$

Therefore $$K | aX + bY = c$$ has a normal distribution with mean and variance

$$\mu_K = \frac{\mu_X + \Omega_{12}\Omega_{22}^{-1}(c - a\mu_X - b\mu_Y) - r_1}{b_1}$$

$$\sigma^2_{K} = \frac{\Omega_{11} - \Omega_{12}\Omega_{22}^{-1}\Omega_{21}}{b_1^2}$$

• Thank you for your answer. I don't understand what is A could you explain more please?
– sam
May 13 at 5:14
• A is a 2x2 matrix made from the numbers 1, 0, a, b, where a and b are the values from the line we want to condition on: aX + bY = c. Because (X,Y)^T is a 2x1 random vector, multiplying A*(X,Y)^T gives us another 2x1 random vector. May 13 at 13:07
• I still can't understand how this relate to my problem. I am looking for the conditional distribution of X,Y given the line and a base for the line but here you are looking for the conditional distribution of X given aX+bY=c
– sam
May 13 at 13:33
• @sam I don't understand what you mean. The conditional distribution of X,Y given aX +bY = c is known from the conditional distribution of X given aX + bY = c, because if we know that aX + bY = c, then we know that Y = c/b - (a/b)X. There isn't exactly a "joint" distribution, because conditioning on the line tells us the exact relationship between X and Y. Knowing X tells us exactly what Y is. May 13 at 14:01
• I am not convinced that this really answers my question, because what I need is to find out that this fraction: $\frac {p_x(r_1+b_1k)p_y(r_2+b_2k)}{\int_{t} p_x(r_1+b_1t)p_y(r_2+b_2t) dt}$ is actually a gaussian distribution
– sam
May 13 at 14:28