The random variable is normally distributed with mean zero and unit variance. Find the probability that the quadratic equation
has real roots.
Given that the two roots and are real, ﬁnd, giving your answers to three signiﬁcant ﬁgures:
- the probability that both and are greater than ;
- the expected value of .
It is quite difficult to ﬁnd statistics questions at this level that are not too difficult and are also not simple applications of standard methods. For example, tests are not really suitable, because the theory is sophisticated while the applications are usually rather straightforward. Most questions in the probability/statistics area tend therefore to concentrate on probability, and many of these have a bit of pure mathematics thrown in.
Here, the random variable is the coefficient of a quadratic equation, which is rather pleasing. But you have to handle the inequalities carefully. The difficulty is increased by the conditional element: for parts (i) and (ii) you are only interested in the case of real roots.
If you don’t have statistical tables handy, don’t bother to ﬁnd some: just leave the answers in terms of the probability, , that a standard (, ) normally distributed random variable satisﬁes .
Solution to problem 75
The solution of the quadratic is
which has real roots if . Let be the probability that a standard normally distributed variable satisﬁes . Then the probability that is (taking the two tails of the normal distribution)
(i) We need the smaller root to be greater than . The smaller root is . Now provided is real, we have
However, if , then the condition that is real, i.e. , implies the stronger condition . The condition that both roots are real and greater than is therefore
and the probability that both roots are real and greater than is
The conditional probability that both roots are greater than given that they are real is
(ii) The sum of the roots is , so we want the expectation of given that , which is