Question

Can a probability density ever be negative?

Answer 1

Here in this question we have been asked if a probability density can ever be negative. For answering this question we will define probability density and consider an example and evaluate its possible values.

Complete step-by-step solution:
Now considering from the question we have been asked if a probability density can ever be negative.
From the basic concepts of probability we know that a probability density of a continuous random variable $X$ can be defined as derivative of the cumulative distribution function that is $f\left( x \right)=\dfrac{d}{dx}F\left( x \right)$ .
The probability density of a discrete random variable $X$ is defined as a list of each possible value of $X$ together with the probability that $X$ takes that value in one trial of the experiment.
The cumulative distribution function always lies between zero and one.
For example if we had considered an example of a rolling dice, the probability of getting a number greater than $3$ is $\dfrac{1}{2}$. The probability density function will be given as $1\left( \dfrac{1}{6} \right)+2\left( \dfrac{1}{6} \right)+3\left( \dfrac{1}{6} \right)+4\left( \dfrac{1}{6} \right)+5\left( \dfrac{1}{6} \right)+6\left( \dfrac{1}{6} \right)$ which can be simplified and given as

$\begin{aligned} & \Rightarrow \dfrac{1}{6}\left( 1+2+3+4+5+6 \right) \\\ & \Rightarrow \dfrac{21}{6}=\dfrac{7}{2} \\\ \end{aligned}$ .
Therefore we can conclude that the probability density can never be negative.

Note: This is a purely theory based question it can be directly answered from the concepts of probability in a short span of time. Hence we have to be very clear with the concepts of probability for answering questions of this type. Very few mistakes are possible in questions of this type. The reason may be some kind of misconception in the concepts.