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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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    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).2012-12-03

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$X$ 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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    No, two random variables are "almost surely equal" if $P(X \ne Y) = 0$, but they are not necessarily equal: $X=Y$. There is a difference between equality and almost sure equality. Your answer is consistent with some quick searches I just did, but $X \ge 0$ is not at all the same thing as $P(X\ge 0)=1$.2016-11-02
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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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    I am super curious why this solution might warrant a downvote.2012-12-03
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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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    I know what essential range is. All I'm saying is that in the first definition, you're saying$X$is almost surely nonnegative, and in the second it looks like (depending on your definition of "range" — whether you mean essential range or just plain range) you're saying that$X$is always nonnegative. I also don't see why a "nonnegative random variable" is merely almost surely nonnegative rather than always nonnegative.2016-11-02