Determine whether the function is a probability density function over the given interval.
-$$f ( x ) = 9 ^ { x } \text { over } \left[ 0 , \frac { \ln ( 1 + \ln 9 ) } { \ln 9 } \right]$$

Question

Quizplus · Accepted Answer

To check if a function is a probability density function (PDF), we need to verify two conditions:

1. The function must be non-negative for all values in the given interval.
2. The area under the curve (AUC) of the function over the interval must be equal to 1.

Let's verify these conditions for the given function f(x):

1. We have f(x) = $9^x$, which is always positive for any value of x. Moreover, the interval is $0 \leq x \leq \frac{\ln(1+\ln9)}{\ln9}$, which is also positive. Therefore, f(x) is non-negative on the interval.

2. To find the AUC, we need to integrate f(x) over the interval:

$$\int_0^{\frac{\ln(1+\ln9)}{\ln9}} 9^x dx = \left[\frac{1}{\ln9}9^x\right]_0^{\frac{\ln(1+\ln9)}{\ln9}} = \frac{1}{\ln9}(9^{\frac{\ln(1+\ln9)}{\ln9}}-1)$$

This AUC turns out to be 1, as can be verified by using the property $a^{\log_a(b)}=b$. Therefore, f(x) satisfies both conditions and is a valid probability density function over the given interval.

Determine Whether the Function Is a Probability Density Function Over f(x)=9x over [0,ln⁡(1+ln⁡9)ln⁡9]f ( x ) = 9 ^ { x } \text { over } \left[ 0 , \frac { \ln ( 1 + \ln 9 ) } { \ln 9 } \right]f(x)=9x over [0,ln9ln(1+ln9)​]

Determine Whether the Function Is a Probability Density Function Over $f ( x ) = 9 ^ { x } \text { over } \left[ 0 , \frac { \ln ( 1 + \ln 9 ) } { \ln 9 } \right]$