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How would you define a random variable to be non-negative ???

What are some examples of a Negative random variable ???

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A good example of a negative random random variable is my stock portfolio. – copper.hat Dec 3 '12 at 18:08
Think of a random variable as a measurement of some sorts. Some measurements are always positive (eg, the number shown when you throw a die), some are always negative (eg, actual car speed less the speed shown on your properly functioning speedometer, actually this example is non-positive), some are neither (distance walked today less the distance walked yesterday). – copper.hat Dec 3 '12 at 18:15
up vote 6 down vote accepted

$X$ is non-negative just means that $P(X<0)=0$. The opposite of "non-negative" is not "negative," just that the random variable might take a negative value, that is $P(X<0)>0$.

A "negative" random variable is one that is always negative - that is: $P(X<0)=1$. Similarly, for "positive," $P(X>0)=1$. Note that a positive random variable is necessarily non-negative. But a non-negative random variable can be zero.

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so a normally distributed random variable is not non-negative then ??? – user1769197 Dec 3 '12 at 18:10
Definitely, no normal distribution is non-negative. – Thomas Andrews Dec 3 '12 at 18:12
@user1769197 To use less negations, that is: normal distribution on the real numbers has to yield negative values. This is because the bell shape tapers off forever in both directions, including the negative direction. – rschwieb Dec 3 '12 at 18:14

A non-negative random variable is one which takes values greater than or equal to zero with probability one, i.e., $X$ is non-negative if $\mathbb{P}(X \geq 0) = 1$.

A negative random variable is one which takes values less than zero with probability one, i.e., $Y$ is negative if $P(Y < 0) = 1$. An example would a random variable which is equal to $-1$ with probability $1/2$ and equal to $-6$ with probability $1/2$, or if $Y \sim \operatorname{Exponential}(\lambda)$ then $-Y$ is a negative random variable (since $Y$ is a positive random variable).

Note in particular that saying a random variable is non-negative is not the opposite of saying it is negative.

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Suppose your random variable is your net return in dollars on a game in a casino.

If you pay money to play and lose it all (or lose part of it) the variable would be negative.

If you win more than you bet, your return will be positive.

Conceivably, if the game is rigged for you to always lose, all of the possible (nonzero probability) outcomes could result in you losing money. That could be called a "negative random variable".

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And thus, to answer the OP's other question, if there's zero chance that you walk away with less money than you come with, the random variable is non-negative. – Brett Frankel Dec 3 '12 at 18:15
I am super curious why this solution might warrant a downvote. – rschwieb Dec 3 '12 at 18:19
+1 Seems like a good example to me. – copper.hat Dec 3 '12 at 18:22

A random variable $X$ is non-negative precisely if $$\Pr(X\ge0)=1.$$

The number of times you're struck by lightning this afternoon is an example.

The time you have to wait for the bus is another.

Viewing $X$ as a function whose domain is a probability space, it means the range of the function is $[0,\infty)$, or sometimes $[0,\infty]$.

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