# Function cannot be used as p.m.f

I have the following question: Give one reason why the following function cannot be a probability mass function:

$$p(x)= \frac{1}{10} (x+2)$$ where $x=1,2,3$

The solution that i have, but i'm not sure if its an answer, is that if I sum all possible outcomes, they are different than $1$.

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It will help you tremendously in your future studies if you learn to write more correctly; it will help in avoiding confusion when you get to more complicated topics.  You are not summing all possible outcomes: the outcomes in this instances are $1, 2, 3$, and whether the outcomes sum to $1$ or not is totally irrelevant. What you are summing is all the probability masses as given by the alleged probability mass function, and these should sum to $1$; but they do not, and hence the given $p(x)$ is not a valid probability mass function. – Dilip Sarwate Nov 11 '12 at 14:36

## 2 Answers

You do need the total probability for all $x$ to sum to $1$, but in this case the sum of the probabilities of all possible outcomes is greater than $1$, not less than.

That is, $3/10 + 4/10 + 5/10 = 12/10 > 1$.

So the core reason is right (i.e., sum must be $1$) but your arithmetic looks to be a bit off.

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The sum is $12/10$ which is greater than $1$.

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yup just corrected my mistake, thanks =) – VP. Nov 11 '12 at 10:32
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