### Video Transcript

The function in the given table is a probability function of a discrete random variable 𝑋. Given the expected value of 𝑋 is four, find the values of 𝑎 and 𝑏. When 𝑥 equals one, 𝑓 of 𝑥 equals 0.2. When 𝑥 equals three, 𝑓 of 𝑥 equals 0.2. When 𝑥 equals 𝑏, 𝑓 of 𝑥 equals 𝑎. When 𝑥 equals five, 𝑓 of 𝑥 is equal to 0.2. And when 𝑥 equals six, 𝑓 of 𝑥 equals 0.3.

As we are dealing with a probability function, we know that the sum of the 𝑓 of 𝑥 column is equal to one. The probabilities add up to one. The sum of 𝑓 of 𝑥 is 0.2 plus 0.2 plus 𝑎 plus 0.2 plus 0.3. This must be equal to one. Grouping the numbers gives us 𝑎 plus 0.9 equals one. And finally, subtracting 0.9 from both sides of the equation gives us a value of 𝑎 of 0.1.

In order to calculate the value of 𝑏, we need to use the fact that the 𝐸 of 𝑥, the expected value, is equal to the sum of 𝑥 multiplied by 𝑓 of 𝑥. In this case, the answer of the expected value is four in this question one multiplied by 0.2 plus three multiplied by 0.2 plus 𝑏 multiplied by 0.1 plus five multiplied by 0.2 plus six multiplied by 0.3 is equal to four.

Simplifying this equation gives us 3.6 plus 0.1𝑏 is equal to four. This means that 0.1𝑏 is equal to 0.4. And finally, dividing both sides of this equation by 0.1 gives us 𝑏 is equal to four. Therefore, for the function in the table with an expected value of four, 𝑎 is equal to 0.1 and 𝑏 is equal to four.