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Chapter 2: Problem 3

Let \(p\left(x_{1}, x_{2}\right)=\frac{1}{16}, x_{1}=1,2,3,4\), and \(x_{2}=1,2,3,4\), zero elsewhere, be the joint pmf of \(X_{1}\) and \(X_{2}\). Show that \(X_{1}\) and \(X_{2}\) are independent.

Short Answer

Expert verified
Yes, \(X_1\) and \(X_2\) are independent because their joint probability mass function can be expressed as a product of their individual probability mass functions.

Step by step solution

01

Compute Individual pmfs

First, calculate the individual probability mass functions \(p(x_1)\) and \(p(x_2)\) by summing the joint pmf over all possible values of the other variable. In this case, \(p(x_1) = \sum_{x_2=1}^{4} p(x_1, x_2)\) and \(p(x_2) = \sum_{x_1=1}^{4} p(x_1, x_2)\). Since the joint pmf \(p(x_1, x_2)\) is the same for all \(x_1\) and \(x_2\), then \(p(x_1) = \frac{1}{4}\) and \(p(x_2) = \frac{1}{4}\) for all \(x_1\) and \(x_2\).
02

Compare Product of Individual pmfs with Joint pmf

Now, verify if \(p(x_1, x_2) = p(x_1) \cdot p(x_2)\) for each combination of \(x_1\) and \(x_2\). Substituting the values yield: \(\frac{1}{16} = \frac{1}{4} \cdot \frac{1}{4}\) which indeed holds. Thus, the joint pmf can be expressed as the product of the individual pmfs.
03

Conclude

Since the joint pmf is equal to the product of the individual pmfs for all \(x_1\) and \(x_2\), you can conclude that the random variables \(X_1\) and \(X_2\) are independent.

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Most popular questions from this chapter

Show that the function \(F(x, y)\) that is equal to 1 provided that \(x+2 y \geq 1\), and that is equal to zero provided that \(x+2 y<1\), cannot be a distribution function of two random variables.

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Let the random variables \(X_{1}\) and \(X_{2}\) have the joint pmf described as follows: $$\begin{array}{c|cccccc}\left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (0,2) & (1,0) & (1,1) & (1,2) \\ \hline f\left(x_{1}, x_{2}\right) & \frac{2}{12} & \frac{3}{12} & \frac{2}{12} & \frac{2}{12} & \frac{2}{12} & \frac{1}{12} \end{array}$$ and \(f\left(x_{1}, x_{2}\right)\) is equal to zero elsewhere. (a) Write these probabilities in a rectangular array as in Example 2.1.3, recording each marginal pdf in the "margins". (b) What is \(P\left(X_{1}+X_{2}=1\right) ?\)
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