Ceng 272 Statistical Computations
Midterm
Apr 06, 2011 14:40 - 16:30
Good Luck!

Answer all the questions.
Write the solutions explicitly and use the statistical terminology

1. (5 pts) A rigged dice is known to have probability $1/2$ for the outcome 6 and all other outcomes are known to be equally likely. What is the probability for outcome 2?
2. (10 pts) Suppose we have a group of 5 candidates

   i

   Find the number of ways selecting 3 council members.

   ii

   Find the number of ways selecting chair, vice chair, and treasurer from the group of 5 candidates.

3. (20 pts) Given that
   • A: College graduate. B: Smoker. C: Heartdisease.
   • $P(A)=0.7. ~P(B)=0.1.~ P(C)=0.05$
   • $P(A \cap C)=0.035~P(B \cap C)=0.03$
   Find that

   iii

   $P(C\vert A)=?$. Are these two events ($A,C$) independent?

   iv

   $P(C\vert B)=?$. Are these two events ($B,C$) independent?

4. (15 pts) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function
   
   $$f(x)=\left\lbrace \begin{array}{l} \frac{20000}{(x+150)^3},~ x > 0 \\ 0, ~elsewhere \\ \end{array}\right\rbrace$$
   
   
   Find the probability that a bottle of this medicine will have a shell life of

   v

   at least 150 days;

   vi

   anywhere from 60 to 90 days.

5. (15 pts) A private pilot wishes to insure his airplane for $200000. The insurance company estimates that a total loss may occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with probability 0.1. Ignoring all other partial losses, what premium should the insurance company charge each year to realize an average profit of $500?

6. (15 pts) Suppose that the probabilities are 0.4, 0.3, 0.2 and 0.1, respectively, that 0, 1, 2 and 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random variable $X$ representing the number of power failures striking this subdivision.

7. (20 pts) Compute $P(\mu - 2\sigma < X < \mu + 2\sigma)$, where $X$ has the density function
   
   $$f(x)=\left\lbrace \begin{array}{l} 6x(1-x),~~ 0 < x < 1 \\ 0,~~~~~~~~~~~~elsewhere\\ \end{array}\right.$$
   
   
   and compare with the result given in Chebyshev's theorem.