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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.2012-12-03
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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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    so a normally distributed random variable is not non-negative then ???2012-12-03
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    Definitely, no normal distribution is non-negative.2012-12-03
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    @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.2012-12-03
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    So nonnegative means almost surely nonnegative? Can a nonnegative random variable take on negative values (with zero probability, but still, sometimes?) For example, a uniform distribution on $[0,1]$ mapped so that 0 is mapped to $-1$.2016-11-02
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    Now you are getting to mysticism - can a random variable with $P(X=0)=0$ ever be zero? Continuous probabilities are not easy to intuit, and it's actually hard to say if a "random variable" "takes on a value." It usually is good to avoid such terms, and treat cases where $P(X)=0$ as essentially not in the range, for probability purposes. @NeilG2016-11-02
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    Well, either you mean $X\ge 0$ or you mean $P(X\ge 0)$. Which is it? I think a nonnegative random variable is the first — not the second. These distinctions can make a big difference in proofs.2016-11-02
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    Well, what does it mean for $X\geq 0$ if $X$ is a random variable? What do you think a random variable is? That's the question. Two random variables are equal the same if $P(X\neq Y)=0$, for example. For example, the CDF of a random variable will be the same. You've taken an intuition about what a random variable is, and jumped to a very hard question. A random variable in a continuous range *never* takes any single value. There are more cautious ways to intuit continuous rational variables so that you avoid this problem.2016-11-02
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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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    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.2012-12-03
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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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    Those are not equivalent definitions: The first says that the support is $X\ge 0$; the second says that the range is $X \ge 0$. But the support and the range don't always coincide.2016-11-02
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    @NeilG : The "essential range" of $X$ is a subset of $[0,\infty]$ if and only if $\Pr(X\ge0)=1$ and also if and only if the support is a subset of $[0,\infty]$. One could also say that $X$ is equivalent to some random variable $Y$, in the sense that $\Pr(X=Y)=1$, for which the range of $Y$ is a subset of $[0,\infty]$. And I suspect that in a future era that understands mathematics better than we do, these distinctions won't be necessary. However, the distinction between "support" and "range" is another matter: there are _discrete_ random variables whose support is $[0,1]$. $\qquad$2016-11-02
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    (For example, a probability distribution that assigns positive probability to each set $\{q\}$ where $q$ is a rational number between $0$ and $1$ has support $[0,1]$ and range $[0,1]\cap\mathbb Q$.) $\qquad$2016-11-02
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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