Problem 1. Suppose that a continuous random variable X has the probability density function (pdf):
$f(x) = c(x-1)$, $1 \le x \le 2$
0, otherwise
(a) Find the value of a constant $c$ that makes $f(x)$ a legitimate pdf of $x$.
(b) Find the cumulative distribution function (cdf) of $X$. Make sure that your cdf is a function defined for all real numbers.
$F(x) = $
where the bounds are
(c) Compute $P(X > 1.5)$ using the cdf you found in (b).
(d) Find mean and median of $X$. If necessary, round your answer to four decimal places.
Mean:
Median:
Problem 2. Suppose that the number of requests for assistance received by a towing service is a Poisson process with rate $\alpha = 6$ per hour. If necessary, round your answer to four decimal places.
(a) Find mean and standard deviation of the number of requests in 30-minutes.
Mean:
Standard Deviation:
Then compute the probability that there is at most one request in 30-minute interval.
(b) Find mean and standard deviation of the time (in hours) between two successive requests.
Mean:
Standard Deviation:
Then compute the probability that more than 20 minutes elapse between two successive requests.