Question

X is a random variable with probability density function $$ f_X(x) = \begin{cases} e^{-2x}, & x \ge 0 \\ 2e^{4x}, & x < 0 \end{cases} $$ Let $T = X^2$. Determine the probability density function for $T$.

Name: x is a random variable with probability density function fxx begincases e 2x x ge 0 2e4x x 0 endcases let t x2 determine the probability density function for t 82145
Uploaded: 2024-07-30T18:41:24-08:00
Duration: 2 min 12 s
Channel: Adi S
Description: x is a random variable with probability density function fxx begincases e 2x x ge 0 2e4x x 0 endcases let t x2 determine the probability density function for t 82145

          X is a random variable with probability density function
$$
f_X(x) = \begin{cases}
e^{-2x}, & x \ge 0 \\
2e^{4x}, & x < 0
\end{cases}
$$
Let $T = X^2$. Determine the probability density function for $T$.

x is a random variable with probability density function fxx begincases e 2x x ge 0 2e4x x 0 endcases let t x2 determine the probability density function for t 82145

Added by Blanca C.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

07/30/2024

Step 1

We want to find the probability density function $f_T(t)$ for $T$. Since $T = X^2$, we have $X = \pm \sqrt{T}$. For $x \ge 0$, $x = \sqrt{t}$, and for $x < 0$, $x = -\sqrt{t}$. Show more…

Show all steps

Thanks for your feedback!

X is a random variable with probability density function $$ f_X(x) = \begin{cases} e^{-2x}, & x \ge 0 \\ 2e^{4x}, & x < 0 \end{cases} $$ Let $T = X^2$. Determine the probability density function for $T$.