Download Recitation 9(IDIN)

Recitation 9
Idin Motedayen-Aval
Additional Topics:
1. Random variable models: Measurable functions
2. Discrete random variables; conditional probability: chain rule.
Problems:
1. An information source produces symbols at random from a five-letter alphabet:
S  {a, b, c, d , e} . The probabilities of the symbols are ½, ¼, 1/8, and 1/16,
respectively. A data compression system encodes the letters into binary strings as
follows:
a
1
b
01
c
001
d
0001
e
0000
Let Y be the random variable equal to the length of the binary string output by the
system.
a. Find the direct model for Y.
b. Find the functional model for Y.
c. Find the cdf of Y.
2. A continuous random variable has cdf
0
x   2


FX ( x)  c(1  sin( x))   2  x   2

  x
1
2

Find c and compute the pdf of X.
3. The exam grades in a certain class have a Gaussian pdf with mean m and standard
deviation . Find the constants a and b such that the random variable Y=aX+b
has a Gaussian pdf with mean m’ and standard deviation ’.
4. Let Y  e X . Find the cdf and pdf of Y in terms of the cdf and pdf of X. Find the
pdf of Y when X is a Gaussian random variable.
5. Let Y=aX+b, where a and b are constants.
a. Find the characteristic function of Y in terms of the characteristic function
of X.
b. Find the characteristic function of Y if X is a zero-mean, unit-variance
Gaussian random variable.
6. Suppose that light bulbs have exponentially distributed lifetimes with unknown
mean E[ X ] . Suppose we measure the lifetime of n light bulbs, and we estimate
the mean E[ X ] by the arithmetic average Y of the measurements. Apply the
Chebychev inequality to the event {| Y  E[ X ] | a} . What happens as n   ?
Note: Look up the m-Erlang random variable.