WORKSHEET
14 March 2017
Continuous RVs
1.
Let X be a continuous random variable. What is the meaning of pdf? Support of X? What is the
cdf associated with a pdf?
2. Suppose that X is a continuous rv with pdf
(a) What is the value of C?
(b) Find P{X > 1}.
3. The amount of time in hours that a computer functions before breaking down is a continuous
random variable with probability density function given by
What is the probability that (a) a computer will function between 50 and 150 hours before
breaking down? (b) it will function for fewer than 100 hours?
4. The time before recharging of a certain kind of smart phone battery is a random variable having a
probability density function given by
What is the probability that exactly 2 of 5 such batteries in such a smart phone will have to be
replaced within the first 150 hours of operation? (Assume that the events Ei, i = 1, 2, 3, 4, 5, that
the ith such battery will have to be replaced within this time are independent.)
5.
Define expectation of a continuous rv.
6. Find E[X] when the pdf of X is
7.
Explain how a pdf can be obtained from a cdf and vice-versa.
8. Let X ~ U[10, 30] and let Y = X4 . Is Y still uniformly distributed? If not, then what is the pdf of Y ?
What is E[Y]? Find E[Y] without using the cdf technique.
9. What is the pdf of the uniform random variable, X, on the interval [a, b]? Compute its
expectation, variance and mgf.
10. An exponential distribution with mean θ has the pdf
Let X ~ exp(1/4). Let Y = eX.
(a) What is the support of Y ?
b) Use the cdf technique to derive the pdf of Y .
(c) Compute E[Y] using the pdf of Y and then again using the pdf of X .
11.
(a) What is the support of Y?
(b) Use the cdf technique to derive the pdf of Y.
(c) From the result in (b), what is the distribution of Y? Without doing any work, what do you
know E[Y] to be?
12. Let Z ~ N(0, 1) and let Y = Z2 . Derive the pdf of Y using the cdf technique.
Note: we say that Z has a normal distribution with mean 0 and sd 1 if
1 −𝑥 2
𝑓𝑍 (𝑥) =
𝑒 2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑥.
√2𝜋
(a) What is the support of Y?
(b) Use the cdf technique to derive the pdf of Y.
(c) From the result in (b), what is the distribution of Y? Without doing any work, what do you
know E[Y] to be?
The method of the physical sciences is based upon the induction which leads us to
expect the recurrence of a phenomenon when the circumstances which give rise to it
are repeated. If all the circumstances could be simultaneously reproduced, this
principle could be fearlessly applied; but this never happens; some of the
circumstances will always be missing. Are we absolutely certain that they are
unimportant? Evidently not! It may be probable, but it cannot be rigorously certain.
Hence the importance of the rôle that is played in the physical sciences by the law of
probability. The calculus of probabilities is therefore not merely a recreation, or a
guide to the baccarat player; and we must thoroughly examine the principles on
which it is based.
- Henri Poincaré, Science and Hypothesis (1905)