Video: Calculating the Expected Value from of a Discrete Random Variable

The discrete random variable 𝑋 has the shown probability distribution. Find the value of π‘˜. Hence determine the expected value of 𝑋.


Video Transcript

The discrete random variable 𝑋 has the shown probability distribution. Find the value of π‘˜.

A probability distribution is a table of values showing the probabilities of various outcomes of an experiment. We know that the sum of all the probabilities for all possible outcomes of the experiment must be one. By adding together the given probabilities then, we can construct an equation to help us calculate the value of π‘˜.

The sum of the probabilities from our table is given by 0.1 plus 0.3 plus 0.2 plus 0.1 plus 0.1 plus π‘˜. And we know this must be equal to one. So this is our equation. Adding together these decimal values gives us 0.8 plus π‘˜ is equal to one. We can solve this equation to calculate the value of π‘˜ by subtracting 0.8 from both sides. One minus 0.8 is 0.2. So π‘˜ is equal to 0.2.

Hence determine the expected value of 𝑋.

Let’s begin by replacing the value of π‘˜ in our table with 0.2. Now that we have a fully complete table, we can calculate the expected value of 𝑋 by using the formula. It’s given by 𝐸 of 𝑋 is equal to the sum of π‘₯ multiplied by 𝑃 of π‘₯. It’s the sum of each of the possible outcomes multiplied by the probability of that outcome occurring.

In the case of our probability distribution, it’s one multiplied by 0.1 plus two multiplied by 0.3 plus three multiplied by 0.2. We’re now on the fourth column. And that gives us four multiplied by 0.1 plus five multiplied by 0.1 plus six multiplied by π‘˜, which we worked out to be 0.2. We can evaluate each of these products to give us 0.1 plus 0.6 plus 0.6 plus 0.4 plus 0.5 plus 1.2, which is equal to 3.4. We’ve calculated the expected value of 𝑋 then to be 3.4.

We can check whether this is a sensible answer by looking at the values in our table. The possible outcomes for π‘₯ are one through to six. 3.4 is roughly halfway between these. So it’s likely that we’ve calculated this value correctly. We know straightaway that our answer was wrong if we had a solution that was lower than one or higher than six.

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