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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?

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    May 22, 2017 at 16:44
  • $\begingroup$ @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! $\endgroup$
    – HannaMao
    May 22, 2017 at 16:55
  • $\begingroup$ It computes the distribution function of $X$ conditional on an inequality. Of course it uses the joint distribution function to do that, but from what you have stated in your question it appears we are supposed to assume $X$ and $Y$ are independent, when your joint distribution is fully specified. $\endgroup$
    – whuber
    May 22, 2017 at 16:56

1 Answer 1

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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 mimic 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('$P(X|X>Y)$')

enter image description here

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('$P(X|X<Y)$')

enter image description here

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  • $\begingroup$ 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). $\endgroup$
    – HannaMao
    May 23, 2017 at 4:47
  • $\begingroup$ @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? $\endgroup$
    – HannaMao
    May 23, 2017 at 15:34
  • $\begingroup$ I think that in the last line of the proof there should be an absolute value of $x_1$. The expected value of this distribution is not defined since it diverges to infinity, correct? $\endgroup$
    – Mattiatore
    Jan 25, 2023 at 13:04
  • $\begingroup$ Actually the final distribution should have a mean, therefore I think this answer is incorrect and I am suspicious about the statement "is equivalent to a product of a uniform random variable" I don't think there is a uniform distribution involved. $\endgroup$
    – Mattiatore
    Jan 25, 2023 at 13:23

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