The discrete random variable 𝑋 has
the shown probability distribution. Find the value of 𝑘.
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
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 𝑘. And we know this must be equal to
one. So this is our equation. Adding together these decimal
values gives us 0.8 plus 𝑘 is equal to one. We can solve this equation to
calculate the value of 𝑘 by subtracting 0.8 from both sides. One minus 0.8 is 0.2. So 𝑘 is equal to 0.2.
Hence determine the expected value
Let’s begin by replacing the value
of 𝑘 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 𝑥 multiplied by 𝑃 of 𝑥. 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 𝑘, 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 𝑥 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