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Math  340 
Fall 2004

Exam 1

Problem 1.  In each case below give the appropriate probability.

a. Let $X$ be binomial $B(20, \frac{2}{3})$. Calculate $P(X < 2)$.
b. Let $X$ be geometric with $P(S) = \frac{2}{5}$. Calculate $P(X > 2)$.
c. Let $X$ be hypergeometric with $N=25,  n=15,  G=15$.  Calculate $P(X=10)$, i.e., $g=10$.

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Problem 2.  The fraction of the people in a population who have a certain disease is 0.75\%. A diagnostic test is available to test for the disease. But for
a healthy person the chance of being falsely diagnosed as having the disease is 0.5\%, while for someone with the disease the chance of being falsely diagnosed as healthy is
20\%. Suppose the test is performed on a person selected at random from the population.  Assuming the test shows a positive result, what is the probability that the person
tested has the disease?

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Problem 3. 
a.  An experiment consists of rolling a die 100 times and counting the number of occurrences of fours. Use an appropriate normal distribution to approximate the
probability of observing at least 30 fours.

b.  Now suppose that we repeat the experiment described in part (a) above once every day for a year (365 days). Use an appropriate Poisson distribution to
approximate the probability that  we observe 30 or more fours, on a given day, at least three times over the one year period.

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Problem 4.  An urn consists of four numbered balls, e.g., $S = \{1, 2, 3, 4\}$. An experiment consists of drawing two balls from $S$ without
replacement. Let $X_{1}$ and $X_{2}$ represent the outcomes of the first and second draws, respectively.

a.  Give the joint distribution of $(X_{1}, X_{2})$.

b.   Let $Y_{1} = \mbox{max}(X_{1},X_{2})$ and $Y_{2} = \mbox{min}(X_{1},X_{2})$. Give the joint distribution of $(Y_{1},Y_{2})$. Are $Y_{1}$ and
$Y_{2}$ identically distributed? Explain.

c.  Let $W = Y_{1}+Y_{2}$. Give the probability distribution of $W$.

d.Give the conditional distribution $P(Y_{2} | X_{2} = 2)$.

e.  Give the following expectations:  $E(W)$ and $E(Y_{2}|X_{2}=2)$.

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