### Video Transcript

The function in the given table is a probability distribution of a discrete random variable 𝑋. Find the value of 𝑎.

The function in the table is a probability distribution. And so the value of 𝑓 of little 𝑥 is the probability that the discrete random variable takes the value 𝑥. For example, when 𝑥𝑖 is two, 𝑓 of 𝑥𝑖 is seven 𝑎. So the table tells us that 𝑓 of two is seven 𝑎. And because 𝑓 of two is the probability that 𝑥 equals two, we can see that the probability that 𝑥 equals two is seven 𝑎. So if this column tells us that the probability that 𝑥 is equal to two is seven 𝑎, what do the other columns tell us? From the next column, we have that the probability that 𝑥 is three is five 𝑎. From this column, we see that the probability that 𝑥 is equal to four is nine 𝑎. And from the final column, that the probability that 𝑥 is equal to five equals three 𝑎.

So we have all these probabilities in terms of this unknown value 𝑎, and we wanted to find the value of 𝑎. We use the fact that the sum of the probabilities of all the outcomes must be one. We can see that there are four possible outcomes for the random variable 𝑥, namely two, three, four, and five. And the sum of the probabilities of these outcomes must be one.

We can now substitute the expressions we have for these probabilities in terms of 𝑎. And we can combine all the like terms on the left-hand side to get that 24 𝑎 is equal to one. And hence, that the value of 𝑎 is one over 24.

The main fact that we used to get this answer was that the sum of the probabilities of all the outcomes of a discrete random variable is one. We can express this in sigma notation, in writing the sum over 𝑖 of the probability that 𝑥 is equal to 𝑥𝑖 is equal to one.