The function in the given table is a probability function of a discrete random variable 𝑋. Given that the expected value of 𝑋 is 254 over 57, find the value of 𝐵.
There are two things we need to recall here. The first is that the probabilities of all possible outcomes of our discrete random variable sum to one. We can therefore form and solve an equation in terms of 𝑎 to help us evaluate the exact probabilities in our table. Collecting like terms gives us the equation 19𝑎 add a third equals one. Subtracting a third from both sides leaves us with 19𝑎 equals two-thirds. And finally, dividing through by 19 gives us a solution of 𝑎 equals two out of 57.
Let’s substitute this value of 𝑎 back into our table. We now have a fully complete row of probabilities. The second piece of information we now require is that of the expected value of 𝑥. The formula for the expectation of a discrete random variable is given by the sum of all the variables multiplied by the respective probabilities. In this case, that’s represented by 𝑓 of 𝑥 because it’s a probability function.
Once again, we can apply this formula to the new values in our table and form an equation in terms of 𝐵. Simplifying and collecting like terms gives us 140 over 57 add 𝐵 over three equals 254 over 57. Subtracting 140 over 57 from both sides leaves us with 𝐵 over three equals 114 over 57. Finally, multiplying through by three gives us a solution of 𝐵 equals six.
Our final answer is 𝐵 equals six.