Suppose that we wish to predict whether a given stock will issue a dividend this year (“Yes” or “No”) based on X, last year’s percent profit. We examine a large number of companies and discover that the mean value of X for companies that issued a dividend was \(\bar{X} = 10\), while the mean for those that didn’t was \(\bar{X} = 0\). In addition, the variance of X for these two sets of companies was \(\sigma^2 = 36\). Finally, 80% of companies issued dividends. Assuming that X follows a normal distribution, predict the probability that a company will issue a dividend this year given that its percentage profit was X = 4 last year.

Hint: Recall that the density function for a normal random variable is f(x) = √ 1 e−(x−μ)2/2σ2 . You will need to use Bayes’ theorem

Hint: See Question 2

Y = Did company pay dividend

X = last year's percent

X_bar_yes = 10% implies dividend paid
X_bar_no = 0% implies dividend NOT paid
80% of companies issues dividend
sigma^2 = 36

Given percentage profit was X = 4 last year