Lecture 6
Brief Probability Overview
A probability space is a triplet {, , P} in which  is a set,  is a family of subsets of
 , representing the events, and P is a probability measure on  .
 has some closure properties to guarantee that operations between events are contained
in  :
1.   
2. For A  , the complement of A, denoted Ac  
3. For A1 , A2 ,, An ,  , A1  A2    An    
P :   [0,1] satisfying:
1. P()  1
2. For an infinite sequence of mutually disjoint events

A1 , A2 ,, An ,  , Ai  A j  , P( A1  A2    An  )   P( A j ) .
j 1
A random variable (r.v.)on  is a measurable function X :   R . Measurability
means that all events expressed in terms of X belong to the family  . In particular it is
enough to require that {   : X ( )  x}  , for all x  R .
Every random variable has a probability distribution. For a discrete random variable we
define f ( x)  P( : X ( )  x) . For a continuous random variable, we define the
cumulative distribution function F ( x)  P( : X ( )  x) . In most cases, F (x) is smooth
x
enough such that there exists f ( x)  0, integrable , and such that F(x) 
 f(t)dt . f is called

the probability density function of the random variable X.
We define the mean, variance and standard deviation (whenever the quantities exist) of a
random variable by:
  E ( X )   x  f ( x) dx,
R
V ( X )    E  ( X   ) 2    ( x   ) 2 f ( x) dx  E ( X 2 )   2 , if X is continuous, and
2
R
   x  f ( x), and    ( x   ) 2 f ( x)  x 2 f ( x)   2 , if X is discrete.
2
x
x
x
The standard deviation is defined by    .
Example 1. Consider flipping a coin once and observing whether it lands Heads or Tails.
Define the random variable X  1 if the coin lands Heads, and X  0 if the coin lands
Tails. Then, X is a discrete random variable with probability distribution
f (0)  P( X  0)  0.5 , and f (1)  P( X  1)  0.5 .
X is a particular example of a Bernoulli random variable.
Definition 1. A random variable that takes only the values 0 and 1, and for which
P( X  1)  p, 0  p  1 is called a Bernoulli random variable.
2
Example 2. Imagine yourself blindfolded and having to cut a ribbon of length 1 yard
suspended between two posts. Define a random variable to be the length of one of the
two pieces of ribbon after you cut it. This is an example of a continuous random variable.
Assuming that you have equal chance of making the cut at any point on the ribbon, the
probability of cutting a piece of length say d is proportional to d, and since the total
length is one, the probability is exactly d. This is an example of a uniform random
variable. Its probability density function is f ( x)  1, for all x  [0,1] .
If we want to find the probability that the length of the cut piece is less than 0.5 of a yard,
0.5
we compute P( X  0.5) 

f ( x) dx  0.5 . Similarly, the probability that its length is
0
between 0.25 and 0.5 is:
0.5
P(0.25  X  0.5) 

f ( x) dx  0.25
0.25
In general, a uniform random variable on an interval [a, b] has a probability density
1
, for all x  [a, b] .
function given by f ( x) 
ba
A very important example of a continuous random variable is the normal random
variable.
The probability density function of a normal random variable is given by
( x   )2

1
2
f ( x,  ,  ) 
e 2 , x  R ,
2 
where   R, and  >0 .  is the mean of the random variable, and  is the standard
deviation. The figure below shows the changes in the densities of normal variables when
the values of  and  change.
Definition 2. Two events A and B are independent if P( A  B)  P( A)  P( B) .
If X and Y are random variables, we can define their joint probability distribution as
having the density f ( x, y ) satisfying:
f ( x, y )  0,  f ( x, y )dx dy  1, if X and Y are continuous random variables, and
R2
f ( x, y )  0,  f ( x, y )  1, if X and Y are discrete random variables.
x
y
From the joint probability distribution we may derive the probability distributions of the
variables X and Y; we denote them f X ( x), fY ( y ) and we refer to them as the marginal
probability distributions of the variables X and Y, respectively. We have:
f X ( x)   f ( x, y ) dy, fY ( y )   f ( x, y ) dx, if X and Y are continuous r.v., and
R
f X ( x) 
 f ( x, y), f
y
R
Y
( x) 
 f ( x, y), if X and Y are discrete r.v.
x
Definition 3. The random variables X and Y are independent if f ( x, y)  f X ( x)  fY ( y) .
Properties
1. If X and Y are random variables and a and b are real numbers, then
E (aX  bY )  aE ( X )  bE (Y )
The property generalizes to a linear combination of n random variables.
n
n
i 1
i 1
E ( ai X i )   ai E ( X i ) .
2. If X and Y are independent random variables, h1 , h2 are real functions, and a and b are
real numbers, then
E  h1 ( X )h2 (Y )   E  h1 ( X )  E  h2 (Y ) 
V (aX  bY )  a 2V ( X )  b 2V (Y )
Again, the property generalizes to a linear combination of independent random variables:
n
n
i 1
i 1
V ( ai X i )   ai2V ( X i )
3. If X i , i  1,..., n are independent normally distributed random variables, with means i
n
and variances  i2 , and a1 ,..., an are real numbers, then Y   ai X i , is a normal
i 1
n
n
i 1
i 1
random variable with mean    ai i and variance  2   ai i2 .
The Central Limit Theorem
Let X 1 ,..., X n be independent and identically distributed random variables (i.i.d. for short)
all having a distribution with mean  and standard deviation  . Then X 
approximately normal, N (  ,
2
n
2
n
X
n
i
is
) (the notation means, the mean is  and the variance is
.
A graduate version of the Central Limit Theorem is:
Let X1 ,..., X n ,... be a sequence of iid random variables all having mean  and standard
deviation  . Then
n
Sn 
X
i 1
i
 n
 n
converges in distribution to a standard normal variable, N (0, 1) .
And now, here are some exercises to refresh your probability knowledge.
EXERCISES
1. Consider the model d  2, N  1, r  1/10, S (0)  6, S (1, 1 )  11/ 2, S (1, 2 )  77 /10 .
(This was also Exercise 1 in Lecture 3.) Find the expected value of the stock price,
the variance and the standard deviation with respect to the risk-neutral probability
measure.
2. Consider the one step binomial model with d  1  r  u , r being the risk free interest
rate corresponding to one step. Find a formula for the mean and the standard
deviation of the stock price with respect to the risk neutral probability measure.
3. Let Y be a binomial random variable with n trials and probability p of success. Notice
that Y can be seen as a sum of n independent Bernoulli random variables which take
n
the value 1 with probability p, that is
Y   X i , where X i are i.i.d. Bernoulli
i 1
r.v.(think that you add 1 each time you have a success in a trial, so Y counts he
number of successes). Use this representation to derive the formulae for the mean and
variance of the binomial distribution, i.e.,   np,  2  np(1  p) .
4. Repeat Exercise 2 for the two-step binomial model.
5. Find formulae for the mean and variance of the stock price in the binomial model that
has n steps. (Hint: By all means do not do brute force computations, but rather use the
distribution of the stock price.)
6. We define the covariance of two random variables by
Cov( X , Y )  E  ( X   X )(Y  Y ) 
Show that Cov( X , Y )  E ( XY )   X Y .
7. Let X and Y be random variables (I did not say independent) having
E ( X )  19, E (Y )  25,V ( X )  25, V (Y )  144, and  ( X , Y )  1/ 2 .
Cov( X , Y )
Note:  ( X , Y ) 
.  ( X , Y ) is called the coefficient of correlation.
V ( X )V (Y )
Compute the following quantities:
a) Cov( X , Y ) b) Cov( X  Y , X  Y ) c) V ( X  Y ) d) V (3 X  4Y )