The function in the given table is a probability distribution function of a discrete random variable 𝑥. Find the value of 𝑎.
Remember, a discrete random variable can take on multiple different values, each with an associated probability, as long as those values are discrete. Now, the probability distribution function, here that’s represented by a table, generates probabilities of value 𝑓 of 𝑥, given some outcome of value 𝑥. And the following properties must hold.
First, the sum of 𝑓𝑥 must be equal to one. Second, any individual value of 𝑓 of 𝑥, in other words any individual probability given in the table, must be greater than or equal to zero and less than or equal to one. So the property we’re going to apply in order to be able to answer this question is the first of these. It’s the sum of 𝑓𝑥 must be equal to one.
The values of 𝑓 of 𝑥 are given in the second row of our table. Since these must sum to one, we can say that one-fifth plus one-tenth plus three-tenths plus one-tenth plus 𝑎 must be equal to one. Now, of course, one-fifth is equivalent to two-tenths. So the sum of each of these fractions is seven-tenths. And seven-tenths plus 𝑎 is equal to one. To solve for 𝑎, we’ll subtract seven-tenths from both sides, where one minus seven-tenths is equal to three-tenths. So 𝑎 is equal to three-tenths.