7. (12%) A test has probability q of success. X represents the random variable of trial times that first success happens. Please derive E(X) = 1 / q.

8. (14%) P represents the probability of adverse reaction not happening for a vaccination. P is a random variable with density functions:$$\left\{ \begin{array}{ll} 12p^2(1-p) & 0 \le p \le 1 \\ 0 & otherwise \end{array} \right.$$Let n be the number of persons who have been injected. Y is the number of persons that adverse reaction does not happen after vaccination. Please calculate E(Y) if n=10.

7. (12%) A test has probability \( q \) of success. \( X \) represents the random variable of trial times that first success happens. Please derive \( E(X) = 1 / q \).

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