Work out the expected value of the random variable 𝑋 whose probability distribution is shown.
Let’s begin by presenting this information in table form. It’s not necessary though it does make it easier to work out what we need to do. The four possible values for our random variable 𝑥 are one, two, three, and four. The probability that 𝑋 is equal to one is given by the height of this first bar; it’s 0.2. The probability that 𝑋 is equal to two is given by the height of the second bar, which is 0.3. The probability that 𝑋 is equal to three is 0.3. And the probability that 𝑋 is equal to four is 0.2.
We can double-check whether we’ve calculated these probabilities correctly because we know that they should all sum to one: 0.2 plus 0.3 plus 0.3 plus 0.2 does indeed equal one.
Now, let’s recall the formula for the expected value of 𝑋. It’s the sum of each of the possible outcomes multiplied by the probability of this outcome occurring. Let’s substitute then what we have into this formula.
For the first column, 𝑥 multiplied by the probability of 𝑋 is one multiplied by 0.2. For the second column, it’s two multiplied by 0.3. The probability that 𝑋 is equal to three is 0.3. So 𝑥 multiplied by the probability of 𝑋 here is three multiplied by 0.3. And for our fourth and final column, it’s four multiplied by 0.2. Evaluating each of these products, we get 0.2 plus 0.6 plus 0.9 plus 0.8 which is 2.5.
So we have that the expected value of 𝑋 is 2.5. We can look at our table to check whether this answer is likely to be correct. Since the possible values of 𝑥 are one, two, three, and four and 2.5 is halfway between one and four, 2.5 is likely to be correct for the expected value of our probability distribution.