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Math 340\hskip 5.5 in Spring 1994  \\
Exam 2  \\ 
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Name:\ \rule{2.5in}{0.002in}
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Problem 1.  (35 points)

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Consider the function $f(t) = \lambda e^{-\lambda t} I_{[0,\infty)}, \lambda > 0$.

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a.\ \ Show that $f(t)$ is a density function, i.e., $f(t)$ can serve as the density for a random variable $X$.

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b.\ \ Derive the distribution function associated with $f(t)$, i.e., the distribution of $X$.  (Need to show all
of your calculations.)

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c.\ \ Calculate the expectation of $X$.  (Need to show all of your work.)

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d.\ \ Let $\lambda = 2$ and define $A = [0.5, 1.5]$.  Calculate the probability of $A$.

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Problem 2.  (30 points)

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Take $\Omega$ to be a set of 6 real numbers.  

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a.\ \ Define a probability measure, $P$, and a random variable, $X$, on it which takes the values 1, 2, 3,
and 4 with probabilities $\frac{1}{5}, \frac{1}{5}, \frac{1}{2}$, and $\frac{1}{10}$ respectively;
{\em and}, another random variable, $Y$, which takes the values $e$, $\pi$, and $\sqrt{2}$ with
probabilities $\frac{1}{2}, \frac{1}{10}$, and $\frac{4}{10}$ respectively.

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b.\ \ Find the probability distribution of $XY$.

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Problem 3.  (35 points)

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Die $A$ has four red and two white faces, whereas die $B$ has two red and four white faces.  A coin is flipped
{\em once}.  If it falls heads, the game continues by throwing die $A$ alone.  If it falls tails, die $B$ is to be
used.

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a.\ \ Show that the probability of red at any throw is $\frac{1}{2}$.

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b.\ \ If the first two throws resulted in red, what is the probability of red at the third throw?

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c.\ \ If red turns up at the first $n$ throws, what is the probability that die $A$ is being used?

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