Question

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{9 x^{2}}{5}-\frac{1}{10} ;[-1,1]$$

Name: Problem 9, Chapter 11: Calculus with Applications
Uploaded: 2020-05-29T21:17:35-08:00
Duration: 1 min 32 s
Channel: Joshua Sieverding
Description: Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. f(x)=(9 x^2)/(5)-(1)/(10) ;[-1,1]

   Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{9 x^{2}}{5}-\frac{1}{10} ;[-1,1]$$

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 11, Problem 9

Instant Answer

Solved by Expert Joshua Sieverding

05/29/2020

Step 1

For the function $f(x) = \frac{9x^2}{5} - \frac{1}{10}$, we can see that it is a quadratic function with a negative coefficient for the $x^2$ term. This means that the function will be negative for some values of $x$. To check if the function is non-negative on Show more…

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Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{9 x^{2}}{5}-\frac{1}{10} ;[-1,1]$$

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Key Concepts

Probability Density Function

A probability density function (pdf) is a function used to describe the likelihood of a continuous random variable taking on a specific value. For a function to be a valid pdf, it must satisfy several fundamental properties, including being non-negative over its domain and integrating to 1 over the interval of interest.

Non-Negativity Condition

One essential requirement for any function to qualify as a probability density function is that it must be non-negative over the entire interval of interest. This ensures that the probability assigned to any outcome is not negative, as probabilities must range from 0 to 1.

Normalization Condition

The normalization condition for a probability density function demands that the integral of the function over its domain equals 1. This property guarantees that the total probability across all possible outcomes sums to unity, representing a complete probability distribution.

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