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I am shown a table with x and f(x) where f is the probability function of a random variable X.

x    |  1   |   2   |   3   |
f(x) | 1/2  |  1/4  |  1/4  |

What would I need to do to find the distribution function of X from this information?

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up vote 1 down vote accepted

The cumulative distribution function, denoted (capital) $F$, is defind by $F(x)=\Pr(X\le x)$. The usual mistake here is to write $F(1)=1/2$, $F(2)=3/4$, $F(3) = 1$, and think that's it. That's correct as far as it goes, but the domain of $F$ is the entire set of real numbers, not just those in the support of the probability distribution. So, for example, $$ F(2.3) = \Pr(X\le 2.3) = \Pr(X=\text{ either }0\text{ or }1) = \frac34. $$ You need to write a piecewise definition, saying what $F(x)$ is for $x<1$, for $1\le x<2$, for $2\le x<3$, and for $x\ge 3$.

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@rhino18 And be sure to pay careful attention to the difference between $<$ and $\leq$ when you follow the directions in Michael Hardy's last sentence above. – Dilip Sarwate Sep 26 '12 at 17:42

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