BASIC PROBABILITY : HOMEWORK 2
Exercise 1: where does the Poisson distribution come from? (corrected)
Fix > 0. We denote by Xn has the Binomial distribution with parameters
n and /n. Then prove that for any k N we have
P [Xn = k ] P [Y = k ],
n
where Y ha
CONTINUOUS RANDOM VARIABLES
1. Continuous random variable
Denition 1.1. A random variable X is said to be continuous if its distribution function can be written as
x
P [ X x] =
fX (u)du,
for some integrable fX : R [0, ), which is called density function o
EXPECTATION
1. Expectation
Denition 1.1. The expectation, or expected value, or mean value of
X with mass function f is dened by
E [X ] =
xf (x).
x
whenever the sum is absolutely convergent.
Proposition 1.1. If X has mass function f and g : R R, then
E(g
BASIC PROBABILITY : GALTON-WATSON TREES
Here all random variables will have integer values.
Denition 0.1. The probability generating function of the random variable X is dened to be the function GX (s) = E [sX ]. This function is well
dened on the interva
BASIC PROBABILITY : RANDOM VARIABLES
In the following we will consider a probability space (, F , P ). We recall
that: is the sample space and enumerates all possibilities, F is a set of
subsets of , they represent the events, and P is a function on F whi
BASIC PROBABILITY : HOMEWORK 5
Question 1: Show using characteristic functions that the sum of two
independent poisson random variables with parameters and is a poisson
random variable of parameter + .
Question 2: Show that if Xn goes to c R in distributi
BASIC PROBABILITY : HOMEWORK 4
Question 1: Suppose that X has an exponential distribution with parameter 1, and Y = X 2 . Compute the expectation of Y .
Question 2: Let X be a continuous random variable with density distribution function fX and g (x) = 2x
BASIC PROBABILITY : HOMEWORK 3
1. Exercise 1
We call characteristic function of X the function : R C dened by
(t) = E [eitX ].
Question 1: Compute (t) when X is a Bernoulli random variable of
parameter p [0, 1].
Question 2: Compute (t) when X is a Binomia
BASIC PROBABILITY : HOMEWORK 3
Consider a particle on the one-dimensional line, starting at position a Z,
say. This particle performs n random steps as follows. At each time t =
1, 2, . . . , n, we ip a coin. If heads comes up, the particle moves one unit
CONVERGENCE OF RANDOM VARIABLES
1. Characteristic functions
1.1. Denition and basic properties.
Denition 1.1. The characteristic function of X is the function : R
C dened by
(t) = E [eitX ].
The joint characteristic function of X and Y is the function :