# Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $$p_j$$ , and that the answers to the various problems may be considered independent experiments. Let $$S_n$$ be the number of problems that a student will get correct. The professor wishes to choose $$p_j$$ so that $$E(S_n) = 0.7n$$ and so that the variance of $$S_n$$ is as large as possible. Show that, to achieve this, she should choose $$p_j = .7$$ for all j; that is, she should make all the problems have the same difficulty.

I found the expected value and variance of $$S_n$$ below

$$E(S_n) = \sum_{j = 1}^{n}p_j=0.7n$$

$$Var(X_j)=E(X_j^2)-E(X_j)^2=p_j-p_j^2$$

$$Var(S_n)= \sum_{j = 1}^{n}p_j-p_j^2=0.7n-\sum_{j = 1}^{n} p_j^2$$

If this line is reasoning is correct I think I need to next show that $$p_j=0.7$$ for all $$p_j$$ minimises $$\sum_{j = 1}^{n} p_j^2$$, but I'm not sure how.

Note that $$\sum_{j=1}^n p_j\leq \sqrt{n}\sqrt{\sum_{j=1}^n p_j^2}$$ so $$\sum p_j^2\geq\left(\sum_{j=1}^n p_j\right)^2\biggr/n=0.7^2n$$ by the Cauchy Schwarz inequality with equality achieved iff $$p_j=0.7$$ by the equality conditions of Cauchy Schwarz.
• I just read up on the Cauchy Schwartz inequality. Are the two vectors in the inequality $(p_1, p_2,...p_n)$ and $(1,1,1...)$ (with length n)? And they are equal when the $p$ vector is a scalar multiple (0.7x) of the vector of ones? – Yandle May 19 at 20:45
• Yes those are the vectors used for the inequality. The equality conditions of C.S imply that $p_1=\dotsb=p_n=c$ for some $c$. But the condition $\sum p_j=0.7n$ forces $c=0.7$ as desired. – Foobaz John May 19 at 22:42