### Video Transcript

Can the function in the given graph be a probability distribution function?

Remember, a probability distribution function tells us the probability of each event occurring. If the function in the graph is a probability distribution function, then it tells us the probability that ๐ฅ is equal to one, the probability that ๐ฅ is equal to two, and the probability that ๐ฅ is equal to three. So letโs recall what we know about probability distribution functions.

For a function ๐ of ๐ฅ, the sum of all ๐ of ๐ฅ values must be equal to one. In other words, the sum of the probabilities of all possible outcomes is one. It also follows that each individual probability can be no greater than one, but no less than zero. And so to check whether the given graph represents a probability distribution function, weโll look at these two properties.

Weโll begin with the second. Weโll check that each individual probability is in the closed interval zero to one. The first bar reaches a height of one-sixth. So if the function in the graph is a probability distribution function of a discrete random variable ๐ฅ, then it tells us that the probability ๐ฅ is equal to one and, hence, ๐ of one is equal to one-sixth. Similarly, the probability that ๐ฅ is equal to two or ๐ of two is three-sixths. And finally, ๐ of three is two-sixths. We can quite quickly see that each of these individual values of ๐ of ๐ฅ are in the closed interval from zero to one. And so the second property is satisfied.

Weโll now check the first property. In other words, weโll add each of these values and check that they sum to one. Thatโs one-sixth plus three-sixths plus two-sixths. Thatโs equal to six-sixths, which is indeed equal to one. And so the second property is satisfied. The sum of all ๐ of ๐ฅ values is one. And so the function in the given graph can be a probability distribution function. The answer is yes.