Let has the following joint density functionF

Question

Let $\underset{\sim }{X}$ $=\binom{X_1}{X_2}$ has the following joint density function

$f\left ( X_1,X_2 \right )=\left\{\begin{matrix} 4x_1x_2 &,0<x_1<1,0<x_2<1, \\ 0 & ,\textup{otherwise.} \end{matrix}\right.$

Find the marginal distributions, mean vector and variance-covariance matrix. Also, comment on the independence of $X_1$ and $X_2$

02 Apr 2024

Answer :

Word Count : 394

To find the marginal distributions, mean vector, variance, and covariance matrix, let's proceed step by step:

1. Marginal Distributions:

The marginal distributions can be found by integrating the joint density function over the entire range of the other variable.

For $ X_1 $:
\[ f_{X_1}(x_1) = \int_{-\infty}^{\infty} f(x_1, x_2) \, dx_2 = \int_{0}^{1} 4x_1x_2 \, dx_2 = 4x_1\int_{0}^{1} x_2 \, dx_2 = 4x_1\left[\frac{x_2^2}{2}\right]_{0}^{1} = ____ _____ __________ _____ ____.
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$x_1$	$x_2$	$x_3$	$x_4$	$y_1$	$y_2$	$y_3$	$y_4$
10	10	10	8	8.04	9.14	7.46	6.58
8	8	8	8	6.95	8.14	6.77	5.76
13	13	13	8	7.58	8.74	12.74	7.71
9	9	9	8	8.81	8.77	7.11	8.84
11	11	11	8	8.33	9.26	7.81	8.47
14	14	14	8	9.96	8.10	8.84	7.04
6	6	6	8	7.24	6.13	6.08	5.25
4	4	4	19	4.26	3.10	5.39	12.50
12	12	12	8	10.84	9.13	8.15	5.56
7	7	7	8	4.82	7.26	6.42	7.91
5	5	5	8	5.68	4.74	5.73