Distribution of X̄ - Ȳ from Normal Distributions

 2026-05-25

#statistics #normal-distribution #clt #sampling

How is $\bar{X} - \bar{Y}$ normally distributed if X and Y are normally distributed?¶

In [1]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

Let's create two random datasets first, we are going to apply CLT first.

In [2]:

x = np.random.randint(0, 20_000, 5000)
y = np.random.randint(20_000, 100_000, 2000)

In [3]:

x[:10], y[:10]

Out[3]:

(array([15133,  1494,  6679,  3853, 12289, 10007, 11246,  8764,   533,
        17345]),
 array([62035, 73654, 88258, 78436, 57026, 68304, 33468, 96705, 23018,
        89356]))

Our first dataset consists of 5000 integer data points, between 0 and 20.000. And our second dataset consists of 2000 integer data points, between 20.000 and 100.000. They are non-overlapping on purpose, and the purpose is to see the wanted reuslt clearer.

In [4]:

means_x = np.array(
    [np.mean(np.random.choice(x, size=30, replace=False)) for i in range(5000)]
)
means_y = np.array(
    [np.mean(np.random.choice(y, size=30, replace=False)) for i in range(5000)]
)

We sampled 5000 times from each dataset, and the sample size is 30 for the both.

In [5]:

fig, ax = plt.subplots()
sns.kdeplot(means_x, ax=ax, label="x")
sns.kdeplot(means_y, ax=ax, label="y")
ax.legend()
plt.show()

No description has been provided for this image

Because of CLT, we have a two normally distributed data. And the means and the standard deviations are different.

In [6]:

means_minus = means_x - means_y

In [7]:

fig, ax = plt.subplots()
sns.kdeplot(means_x, ax=ax, label="x")
sns.kdeplot(means_y, ax=ax, label="y")
sns.kdeplot(means_minus, ax=ax, label="minus")
ax.legend()
plt.show()

Statistical notation of means_minus = means_x - means_y is $\bar{X} - \bar{Y}$. And as you can see, means_minus is also normally distributed.

We know that:

$Var\[\bar{X} - \bar{Y}\] = Var\[\bar{X}\] + Var\[\bar{Y}\]$
$E\[\bar{X} - \bar{Y}\] = E\[\bar{X}\] - E\[\bar{Y}\]$

Let's prove it.

In [8]:

mean_means_x = np.mean(means_x)
mean_means_y = np.mean(means_y)
var_means_x = np.var(means_x)
var_means_y = np.var(means_y)

In [9]:

np.mean(means_minus), mean_means_x - mean_means_y

Out[9]:

(np.float64(-49645.91423333334), np.float64(-49645.914233333344))

In [10]:

np.var(means_minus), var_means_x + var_means_y

Out[10]:

(np.float64(18194249.373440742), np.float64(18284393.44663439))

Yes, it is indeed correct.

TODO: Mathematically prove the below conclusions:

$Var\[\bar{X} - \bar{Y}\] = Var\[\bar{X}\] + Var\[\bar{Y}\]$
$E\[\bar{X} - \bar{Y}\] = E\[\bar{X}\] - E\[\bar{Y}\]$

TODO: Why do we need X and Y to be normally distributed, doesn't CLT already do that for us?

Proof of Estimators¶

Proof of (X̄ - Ȳ) Estimators

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