Video Transcript
The function in the given table is
a probability function of a discrete random variable 𝑋. Find the value of 𝑎.
Notice that each probability in
this probability distribution function has been expressed in terms of the variable
𝑎 whose value we’ve been asked to find. In order to do this, we need to
recall the key fact that the sum of all the probabilities in a probability
distribution function must be equal to one. We can therefore form an equation
using the four values in the second row of our table. Three 𝑎 plus eight 𝑎 squared plus
four 𝑎 squared plus eight 𝑎 is equal to one. This simplifies to 12𝑎 squared
plus 11𝑎 equals one. And then subtracting one from each
side, we see that we have the quadratic equation 12𝑎 squared plus 11𝑎 minus one is
equal to zero.
This equation can be solved in a
variety of ways. But the easiest method for this
particular quadratic is going to be to solve by factoring. With a little bit of trial and
error or perhaps using factoring by grouping, we see that this quadratic factors as
12𝑎 minus one multiplied by 𝑎 plus one. We then follow the usual method for
solving a quadratic equation by factoring. We set each factor in turn equal to
zero and solve the resulting linear equation, giving two values of 𝑎: 𝑎 equals one
twelfth and 𝑎 equals negative one.
So, we have two possibilities for
the value of 𝑎, both correct roots of this quadratic equation. But only one value makes sense in
the context of this problem. If we look back at our table, we
see, for example, that the probability 𝑋 equals zero is three 𝑎. If we use the value one twelfth,
then this becomes three twelfths or one-quarter. But if we use the value negative
one, this gives negative three. Remember that probabilities must
always be between zero and one, so we can’t have a probability of negative
three. This means that the value 𝑎 equals
negative one, whilst being a correct solution to the quadratic equation, is not
correct in the context of this problem as a value of 𝑎.
We can check whether this value of
one twelfth is correct by calculating all of the probabilities. Three 𝑎 gives three twelfths,
which is equivalent to 36 over 144. And in the same way, we can find
the probabilities as fractions with a denominator of 144 for eight 𝑎 squared, four
𝑎 squared, and eight 𝑎. When we sum these four values
together, we do indeed get 144 over 144 or one, and so this confirms that our value
of 𝑎 is correct. The value of 𝑎 then is one
twelfth.
