Let a random variable $X$ take its values in the set $\{-2, -1, 1, 2, 3\}$. Assume that its pmf function can be written in the form: $$ P(X = x_i)= \begin{cases} \frac{x_i^2}{a},& \text{if } x_i\in\{-2, -1, 1, 2, 3\}\\ 0, & \text{otherwise} \end{cases} $$

Find the constant $a$.

I'm not sure where to even start.

  • 2
    $\begingroup$ If you're not sure where to start, you may want to take a look at some basic properties of PMF: en.wikipedia.org/wiki/Probability_mass_function. Particularly the three properties listed on the wiki page. $\endgroup$
    – Gregory
    Commented Mar 6, 2021 at 23:13
  • $\begingroup$ You want all probabilities of events to be non-negative and the total probability to be $1$ $\endgroup$
    – Henry
    Commented Mar 7, 2021 at 2:22

1 Answer 1


The sum of $x^2/a$ over your set is $19/a$. For this to be a pmf the sum must equal $1$, so $a=19$.


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