Problem 10 Let $X$ have the pmf $$ p(... [FREE SOLUTION]

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Introduction to Mathematical Statistics

Robert V. Hogg, Joeseph McKean, Allen T Craig

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 1: Problem 10

Let $X$ have the pmf $$ p(x)=\left(\frac{1}{2}\right)^{|x|}, \quad x=-1,-2,-3, \ldots $$ Find the pmf of $Y=X^{4}$.

Short Answer

Expert verified

The pmf of $Y$ is given by $p_Y(y)= \left(\frac{1}{2}\right)^{\sqrt[4]{y}}, y = 1, 16, 81, 256, \ldots$ and $0$ otherwise.

Step by step solution

Understanding the Variables

The given random variable $X$ can take the values -1, -2, -3, and so on. The pmf $p(x)$ is given by $\left(\frac{1}{2}\right)^{|x|}$, where $|x|$ is the absolute value of $x$. So, for instance, the probability that $X = -1$ is $\left(\frac{1}{2}\right)^{|-1|}$, or $\frac{1}{2}$, and the probability that $X = -2$ is $\left(\frac{1}{2}\right)^{|-2|}$, or $\frac{1}{4}$, and so on. The new variable $Y$ is defined as $Y = X^4$. Since $X$ can only be negative numbers or zero, and any real number raised to the power of 4 is non-negative, $Y$ can only take on non-negative values that are perfect fourth powers.

Finding the pmf of $Y$

To find the pmf of $Y$, denoted as $p_Y(y)$, after noticing that $Y = X^4$, we are going to see that for each value $x$ of $X$, there correspond values for $y$. Let's take $y = 1$, its corresponding $x$ values are -1 and 1, since $(-1)^4 = 1$ and $1^4 = 1$. However, from the value domain of $X$, we notice $X$ can only be -1. Hence, we have $p_Y(1) = p_X(-1) = \left(\frac{1}{2}\right)^{|-1|} = \frac{1}{2}$. Similarly, for $y = 16$, it corresponding $x$ value can only be -2, hence the probability is $p_Y(16) = p_X(-2) = \left(\frac{1}{2}\right)^{|-2|} = \frac{1}{4}$, and so on. We see that for every $y = x^{4}, y > 0$, there corresponds a unique $x = -\sqrt[4]{y}$.

Putting it all together

To find the pmf of $Y$, one need to consider all values that $Y$ could be and find the corresponding probabilities. So, from the above steps, replace $x = -\sqrt[4]{y}$ to the pmf of $X$, we obtain the probability mass function of $Y$ as $p_Y(y)= \left(\frac{1}{2}\right)^{\sqrt[4]{y}}, y = 1, 16, 81, 256, \ldots$ and $0$ otherwise.

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91影视

Let \(X\) have the pmf $$ p(x)=\left(\frac{1}{2}\right)^{|x|}, \quad x=-1,-2,-3, \ldots $$ Find the pmf of \(Y=X^{4}\).

Short Answer

Step by step solution

Understanding the Variables

Finding the pmf of \(Y\)

Putting it all together

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