Download Probability Theory-Fall 2011 Assignment

Probability Theory-Fall 2011
Assignment-1
1. A, B, C and D were all friends at school. Subsequently each of the six pairs meet up, and
at each such meeting, the pair involved quarrel with some fixed probability (sap p), or
become firm friends with probability (1 − p). Quarrels take place independently of each
other. In future, if any of the four hears a rumor, then she tells it to her firm friends only.
If A hears a rumor, what is the probability that:
(i) D hears it?
(ii) D hears it if A and B have quareled?
(iii) D hears it if she has quarreled with A?
2. A biased coin is tossed repeatedly. Each time there is a probability p of heads. Let pn be
the probability that an even number of heads has occurred after n tosses. Clearly, p0 = 1.
Show that
pn = p(1 − pn−1 ) + (1 − p)pn−1 , if n ≥ 1.
Solve this difference equation.
3. 10% surface of a sphere is colored blue, the rest is red. Show that, irrespective of the
manner in which the colors are distributed, it is possible to inscribe a cube in the sphere
with all its vertices red.
Hint: Use Boole’s inequality. Note that the event that one vertex is red (or blue) cannot,
apriori, be assumed to be independent of the event that some other vertex is red.
4. Independence of discrete random variables. Discrete random variables X and Y
are independent if the events {X = x} and {Y = y} are independent for all x and y.
Equivalently, the random variables are independent if the joint probability mass function
pX,Y (x, y) can be expressed as the product of individual probability mass functions.
Let X1 and X2 be independent random variables which are symmetric1 about zero.
Show that Y = X1 + X2 is a symmetric random variable. Is the result still true if the
assumption of independence is dropped?
5. Out of a population of N animals, n were captured, marked, and released back. Let X
be the number of animals it is necessary to recapture (without re-release i.e., in sampling
language without replacement) in order to obtain m marked animals. Write down an
explicit probability mass function for the random variable X.
6. For a discrete random variable X, its expected value, or the mean, is defined as follows:
X
E(X) =
x P (X = x).
x
Show
P∞ that for a discrete random variable taking non-negative integer values, E(X) =
n=0 P (X > n)
1X is a symmetric (about zero) random variable if P (X = x) = P (X = −x), or equivalently, the probability
mass function of X and −X are the same.
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