# Conditional distribution of a normal distribution given it is smaller/bigger than another normal distribution

Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not.

$P(X|X<Y)$ = ... and

$P(X|X>Y)$ = ...

I am thinking solving this in this way:

\begin{align} P(X|X<Y) &= \frac{P(Y>X|X)P(X)}{P(Y>X)} \\ &= \frac{(1-\Phi(\frac{x-\mu_2}{\sigma_2}))N(\mu_1,\sigma_1)}{\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}\\ P(X|X>Y) &= \frac{P(Y<X|X)P(X)}{P(Y<X)} \\ &= \frac{\Phi(\frac{x-\mu_2}{\sigma_2})N(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})} \end{align}

My questions are:

(1) Whether above solution is correct

(2) How to get the mean and sd for $P(X|X<Y)$ and $P(X|X>Y)$ if they are still normal?

• What do you mean by "truncating" here? What happens to a draw of X if it's less than Y? It seems that you're using the term "truncate" not in its usual statistical meaning – Aksakal May 22 '17 at 15:14
• I am sorry I may have used the wrong term. I would like to get the distribution of X if X < Y and the distribution of X if X> Y. I am wondering if there is any statistical formula can do this. – HannaMao May 22 '17 at 15:20
• I thought the results would be a truncated normal distribution. – HannaMao May 22 '17 at 15:21
• Although the procedure is related to truncation, the distribution function is not truncated. The integral to compute the distribution function of $X$ conditional on $Y\gt X$ is easily reduced to the one computed at stats.stackexchange.com/questions/61080. – whuber May 22 '17 at 16:44
• @whuber If I understand correctly, the referred link calculated the joint distribution of [X, Y>X]. For my question, the conditional distribution should be the joint distribution divided by [Y>X]. Am I right? Thanks! – HannaMao May 22 '17 at 16:55

Whether above solution is correct

Yes.

How to get the mean and sd for $P(X|X<Y)$ and $P(X|X>Y)$ if they are still normal?

They are not normal.

Proof:

Given $P(X | X>Y) = \frac{\Phi(\frac{x-\mu_2}{\sigma_2})\phi_x(\mu_1,\sigma_1)}{1-\Phi(\frac{\mu_2-\mu_1}{\sqrt{\sigma_2^2+\sigma_1^2}})}$ is equivalent to a product of a uniform random variable $(\Phi(\frac{x-\mu_2}{\sigma_2})$ and a normal random variable $(\phi_x(\mu_1,\sigma_1))$

Consider $X_1 \sim N(0, 1)$ and $X_2 \sim U(0,1)$, then the product $Z = X_1X_2$ distrobution is given by:

\begin{align*} F_Z(z) &= P(Z \leq z)\\ &= P(X_1X_2 \leq z)\\ &= \int_{X_1\geq 0}P(X_2 \leq \frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1 +\int_{X_1\leq 0}P(X_2 \geq \frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1\\ &= \int_{X_1\geq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 + \int_{X_1\leq 0}(1-\frac{z}{x_1}) \phi_{X_1}(x_1)\ dx_1\\ &= \frac{1}{2} + \int_{X_1\geq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 - \int_{X_1\leq 0}\frac{z}{x_1} \phi_{X_1}(x_1)\ dx_1 \\ &= \frac{1}{2} + \int\frac{2z}{x_1} \phi_{X_1}(x_1)\ dx_1 \end{align*}

which does not mimick CDF of a normal.

You can however still check if your solution is correct by simulation:

import matplotlib.pyplot as plt
import scipy as sp
import numpy as np

mu1 = 1
sigma1 = 2

mu2 = 2
sigma2 = 3

np.random.seed(42)
X = np.random.normal(mu1, sigma1, 1000)
Y = np.random.normal(mu2, sigma2, 1000)

# P(X|X>Y)
P_X_XgY = X[X>Y]
# P(X|X<Y)
P_X_XlY = X[X<Y]

denom = 1-sp.stats.norm.cdf((mu2-mu1)/np.sqrt(sigma1**2+sigma2**2))

count, bins, ignored = plt.hist(P_X_XgY, 30, normed=True)
plt.plot(bins, 1/(sigma1 * np.sqrt(2 * np.pi)) * \
(sp.stats.norm.cdf((bins-mu2)/sigma2)/denom) *\
np.exp( - (bins - mu1)**2 / (2 * sigma1**2) ), linewidth=2, color='r')
plt.title('$P(X|X>Y)$')


denom = sp.stats.norm.cdf((mu2-mu1)/np.sqrt(sigma1**2+sigma2**2))

count, bins, ignored = plt.hist(P_X_XlY, 30, normed=True)
plt.plot(bins, 1/(sigma1 * np.sqrt(2 * np.pi)) *\
((1-sp.stats.norm.cdf((bins-mu2)/sigma2))/denom) *\
np.exp( - (bins - mu1)**2 / (2 * sigma1**2) ), linewidth=2, color='r')
plt.title('$P(X|X<Y)$')


• This is a wonderful answer! I am wondering if there is a name for the distribution of P(X|X>Y) or P(X|X<Y). – HannaMao May 23 '17 at 4:47
• @whuber I am sorry I still couldn't get it. Here Z is the independent variable for $F_Z(z)$. I wanted it to be discussed more with x being the independent variable, $F_X(x)$. Did I make anything wrong? – HannaMao May 23 '17 at 15:34