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Probability, Random Variables and Density Functions

1. A coin is biased so that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.

2. An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, ¼, ¼, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4pm and 5pm on any sunny Friday. Find the attendant's expected earnings for this particular period.

3. If a dealer's profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function

f(x) = 2(1-x), 0 < x < 1,
0 elsewhere

Find the average profit per automobile.

4. What proportion of individuals can be expected to respond to a certain mail-order solicitation if the proportion X has the density function

f(x) = 2(x+2)/5, 0 < x < 1,
0 elsewhere

5. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2 or 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.

6. The proportion of people who respond to a certain mail-order solicitation is a random variable X having the density function given in #4. Find the variance of X.

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