Let X be a random variable that follows the beta distribution. This random variable is continuous and is defined over the interval from 0 to 1. The probability density function is given by f(x)= x-1(1-x)-1 (a-1)!(B-1)! (a + B-1)! where a andare integers, whose values determine the shape of the probability density function. Because X varies between 0 and 1, we can think of X as the probability that some event (say) E occurs or the proportion of times an event occurs in some population. For example, E could denote the event that a critical part in a newly designed car will lead to a catastrophic failure in accidents at high speeds. The expected value (i.e., mean) of this random variable is [a /(a + ß)]. That is, E(X)= α a+ß The Excel commands for the beta random variable are -beta.dist(x, a, B ,true,0,1) for the cumulative probability distribution, and -beta.dist(x, a, ß,false,0,1) for the probability density function. (a) Now, think in Bayesian terms. Previously P(B) denoted prior probability; and P(B) was a single number between 0 and 1, such as, for example, P(B) = .7. Now, the prior is an entire probability density function f(x). For example, f(x) is the prior probability density for the event E described above. Assume that a =B=2. Show that this Bayesian prior f(x) is a quadratic. Also, graph this function in Excel as follows: In Column A, set x = 0, and then increment x by .01 in successive rows until you reach x = 1.0. That is, the values in Column A are 0, .01, .02, ...., .99, 1.00. In Column B, calculate the corresponding values of f(x). Draw a graph in Excel with x on the horizontal axis and f(x) on the vertical axis. This prior probability density function represents your beliefs (for whatever reasons) about the probability of event E occurring. At what value of x is the quadratic maximized [which is the expected value of X because of the symmetry of f(x)]? You conduct n tests and find that the event E occurs in 1₁ of these n tests. It turns out that the posterior probability density function also has a beta distribution. [In Bayesian theory parlance, we say that the beta distribution is a conjugate prior when the likelihood function is binomial.] The new Excel commands for the posterior density become =beta.dist(x, a +n₁, B+n-n₁ ,true,0,1) for the cumulative probability distribution, and =beta.dist(x, a + n₁, B+n- n₁ ,false,0,1) for the probability density function. Suppose your n = 10 tests and in ₁ = 2 cases the event E occurs (still keeping a =B=2). [These tests are expensive because they involve intentionally crashing cars at high speeds, which, of course, destroys the test cars in order to determine whether there will be a critical failure of the part being tested.] In Column C, calculate the values of the posterior probability density function for x = 0, .01, .02, ...., .99, 1.00 and graph it on the same chart as you graphed the prior probability density function. (b) Using the prior distribution, what is the probability that the proportion of times the event E occurs is between 40% and 60%? Using the posterior distribution, what is the probability that the proportion of times the event E occurs is between 40% and 60%?