1. RANDOM VARIABLES, EXPECTATIONS, VARIANCES ETC.

2. Variable
Recall:
Variable: A characteristic of population or sample that is of interest for us.
Random variable: A function defined on the sample space S that associates a real number with each outcome in S.

3. Random Variables
If X is a function that assigns a real numbered value to every possible event in a sample space of interest, X is called a random variable.
It is denoted by capital letters such as X, Y and Z.
The specified value of the random variable is unknown until the experimental outcome is observed.

4. EXAMPLES
The experiment of flipping a fair coin. Outcome of the flip is a random variable.
S={H,T}X(H)=1 and X(T)=0
Select a student at random from all registered students at METU. We want to know the weight of these students.
X = the weight of the selected student
S: {x: 45kg  X  300kg}

5. Discrete and Continuous Random Variables
A random variable is discrete if it can assume a countable number of values.
A random variable is continuous if it can assume an uncountable number of values.
Discrete random variable
Continuous random variable
After the first value is defined
the second value, and any value
thereafter are known.
After the first value is defined,
any number can be the next one 1 2
3 ...
1/16
1/4
1/2 1
Therefore, the number of
values is countable
Therefore, the number of
values is uncountable

6. DISCRETE RANDOM VARIABLES
If the set of all possible values of a r.v. X is a countable set, then X is called discrete r.v.
The function f(x)=P(X=x) for x=x1,x2, … that assigns the probability to each value x is called the probability mass function (p.m.f.)

7. Requirements for a Discrete Distribution
If a random variable can assume values xi, then the following must be true:

8. EXAMPLE Consider an experiment in which a fair coin is tossed 3 times.
X = The number of heads
Let’s assign 1 for head and 0 for tail. The sample space is
S: {TTT,TTH,THT,HTT,THH,HTH,HHT,HHH}
Possible values of X is 0, 1, 2, 3. Then, the probability distribution of X is x 1 2 3
Total
p(x)
1/8
3/8

9. Example Discrete Uniform distribution:
Example: throw a fair die. P(X=1)=…=P(X=6)=1/6

10. The Cumulative Distribution Function
If X is a random variable, then the cumulative distribution function (cdf), denoted by F(x) is given by
for all real numbers x. It is a non-decreasing step function of x and it is right continuous.

11. The Cumulative Distribution Function
For any two numbers a and b, a  b
P (a  X  b) = F (b) – F (a-)
where a- represents the largest possible X value which is less than a.
If a and b are integers,
P (a  X  b) = F (b) – F(a-1)
Taking a=b,
P( X=a ) = F (a)-F (a-1)

12. EXAMPLE
Let X is the number of days of sick leave taken by a randomly selected employee of a large company during a particular year. If the max. number of allowable sick days per year is 14, possible values of X are 0,1,2,…,14. With F(0)=0.58, F(1)=0.72, F(2)=0.76,F(3)=0.81,F(4)=0.88 and F(5)=0.94, find
P(2  X  5) =
P(X = 3) =
P(X  2) =

13. CONTINUOUS RANDOM VARIABLES
When sample space is uncountable (continuous)
We should not ask for the probability that X is exactly a single number (since that probability is zero). Instead, we need to think about the probability that x is close to a single number.
Example: Continuous Uniform(a,b)

14. Probability Density Function (pdf)
If the probability density around a point x is large, that means the random variable X is likely to be close to x. If, on the other hand, f(x)=0 in some interval, then X won't be in that interval.
Probabilities for continuous distributions are measured over ranges of values rather than single points. A probability indicates the likelihood that a value will fall within an interval.

15. Probability Density Function (pdf)

16. The Cumulative Distribution Function
If X is a random variable, then the cumulative distribution function (cdf), denoted by F(x) is given by
for all real numbers x. It is a non-decreasing function of x and it is right continuous.

17. Example
Let

18. JOINT DISCRETE DISTRIBUTIONS
JOINT DISTRIBUTIONS
In many applications there are more than one random variables of interest, say X1, X2,…,Xk.
JOINT DISCRETE DISTRIBUTIONS
The joint probability mass function (joint pmf) of the k-dimensional discrete rv
X=(X1, X2,…,Xk) is

19. JOINT DISCRETE DISTRIBUTIONS
A function f(x1, x2,…, xk) is the joint pmf for some vector valued rv X=(X1, X2,…,Xk) iff the following properties are satisfied:
f(x1, x2,…, xk) 0 for all (x1, x2,…, xk)
and

20. Example Tossing two fair dice  36 possible sample points
Let X: sum of the two dice;
Y: |difference of the two dice|
For e.g.:
For (3,3), X=6 and Y=0.
For both (4,1) and (1,4), X=5, Y=3.

21. Example Joint pmf of (x,y) x y 2 3 4 5 6 7 8 9 10 11 12 1/36 1 1/18
1/36 1
1/18
Empty cells are equal to 0.
e.g. P(X=7,Y≤4)=f(7,0)+f(7,1)+f(7,2)+f(7,3)+f(7,4)=0+1/18+0+1/18+0=1/9

22. MARGINAL DISCRETE DISTRIBUTIONS
If the pair (X1,X2) of discrete random variables has the joint pmf f(x1,x2), then the marginal pmfs of X1 and X2 are

23. Example
In the previous example,

24. JOINT DISCRETE DISTRIBUTIONS
JOINT CDF:
F(x1,x2) is a cdf iff

25. JOINT CONTINUOUS DISTRIBUTIONS
A k-dimensional vector valued rv X=(X1, X2,…,Xk) is said to be continuous if there is a function f(x1, x2,…, xk), called the joint probability density function (joint pdf), of X, such that the joint cdf can be given as

26. JOINT CONTINUOUS DISTRIBUTIONS
A function f(x1, x2,…, xk) is the joint pdf for some vector valued rv X=(X1, X2,…,Xk) iff the following properties are satisfied:
f(x1, x2,…, xk) 0 for all (x1, x2,…, xk)
and

27. JOINT CONTINUOUS DISTRIBUTIONS
If the pair (X1,X2) of discrete random variables has the joint pdf f(x1,x2), then the marginal pdfs of X1 and X2 are

28. JOINT DISTRIBUTIONS
If X1, X2,…,Xk are independent from each other, then the joint pdf can be given as
And the joint cdf can be written as

29. CONDITIONAL DISTRIBUTIONS
If X1 and X2 are discrete or continuous random variables with joint pdf f(x1,x2), then the conditional pdf of X2 given X1=x1 is defined by
For independent rvs,

30. % promoted (number of employees)
Example
Statistical Analysis of Employment Discrimination Data (Example from Dudewicz & Mishra, 1988; data from Dawson, Hankey and Myers, 1982)
% promoted (number of employees)
Pay grade
Affected class
others 5
100 (6)
84 (80) 7
88 (8)
87 (195) 9
93 (29)
88 (335) 10
7 (102)
8 (695) 11
7 (15)
11 (185) 12
10 (10)
7 (165) 13
0 (2)
9 (81) 14
0 (1)
7 (41)
Affected class might be a minority group or e.g. women

31. Example, cont.
Does this data indicate discrimination against the affected class in promotions in this company?
Let X=(X1,X2,X3) where X1 is pay grade of an employee; X2 is an indicator of whether the employee is in the affected class or not; X3 is an indicator of whether the employee was promoted or not
x1={5,7,9,10,11,12,13,14}; x2={0,1}; x3={0,1}

32. Example, cont.
Pay grade
Affected class
others 10
7 (102)
8 (695)
E.g., in pay grade 10 of this occupation (X1=10) there were 102 members of the affected class and 695 members of the other classes. Seven percent of the affected class in pay grade 10 had been promoted, that is (102)(0.07)=7 individuals out of 102 had been promoted.
Out of 1950 employees, only 173 are in the affected class; this is not atypical in such studies.

33. Example, cont.
Pay grade
Affected class
others 10
7 (102)
8 (695)
E.g. probability of a randomly selected employee being in pay grade 10, being in the affected class, and promoted: P(X1=10,X2=1,X3=1)=7/1950= (Probability function of a discrete 3 dimensional r.v.)
E.g. probability of a randomly selected employee being in pay grade 10 and promoted:
P(X1=10, X3=1)= (7+56)/1950= (Note: 8% of 695 -> 56) (marginal probability function of X1 and X3)

34. Example, cont.
E.g. probability that an employee is in the other class (X2=0) given that the employee is in pay grade 10 (X1=10) and was promoted (X3=1):
P(X2=0| X1=10, X3=1)= P(X1=10,X2=0,X3=1)/P(X1=10, X3=1)
=(56/1950)/(63/1950)=0.89 (conditional probability)
probability that an employee is in the affected class (X2=1) given that the employee is in pay grade 10 (X1=10) and was promoted (X3=1):
P(X2=1| X1=10, X3=1)=(7/1950)/(63/1950)=0.11

35. Describing the Population
We’re interested in describing the population by computing various parameters.
For instance, we calculate the population mean and population variance.

36. EXPECTED VALUES
Let X be a rv with pdf fX(x) and g(X) be a function of X. Then, the expected value (or the mean or the mathematical expectation) of g(X)
providing the sum or the integral exists, i.e.,
<E[g(X)]<.

37. EXPECTED VALUES
E[g(X)] is finite if E[| g(X) |] is finite.

38. Population Mean (Expected Value)
Given a discrete random variable X with values xi, that occur with probabilities p(xi), the population mean of X is

39. Population Variance
Let X be a discrete random variable with possible values xi that occur with probabilities p(xi), and let E(xi) =. The variance of X is defined by
Unit*Unit
Unit

40. EXPECTED VALUE
The expected value or mean value of a continuous random variable X with pdf f(x) is
The variance of a continuous random
variable X with pdf f(x) is

41. EXAMPLE
The pmf for the number of defective items in a lot is as follows
Find the expected number and the variance of
defective items.
Results: E(X)=0.99, Var(X)=0.8699

42. EXAMPLE Let X be a random variable. Its pdf is
f(x)=2(1-x), 0< x < 1
Find E(X) and Var(X).

43. EXAMPLE
What is the mathematical expectation if we win $10 when a die comes up 1 or 6 and lose $5 when it comes up 2, 3, 4 and 5?
X = amount of profit

44. EXAMPLE
A grab-bay contains 6 packages worth $2 each, 11 packages worth $3, and 8 packages worth $4 each. Is it reasonable to pay $3.5 for the option of selecting one of these packages at random?
X = worth of packages

45. EXAMPLE
Let X be a random variable and it is a life length of light bulb. Its pdf is
f(x)=2(1-x), 0< x < 1
Find E(X) and Var(X).

46. Laws of Expected Value
Let X be a rv and a, b, and c be constants. Then, for any two functions g1(x) and g2(x) whose expectations exist,

47. Laws of Expected Value and Variance
Let X be a rv and c be a constant.
Laws of Expected Value
E(c) = c
E(X + c) = E(X) + c
E(cX) = cE(X)
Laws of Variance
V(c) = 0
V(X + c) = V(X)
V(cX) = c2V(X)

48. EXPECTED VALUE If X and Y are independent,
The covariance of X and Y is defined as

49. If (X,Y)~Normal, then X and Y are independent iff
EXPECTED VALUE
If X and Y are independent,
The reverse is usually not correct! It is only correct under normal distribution.
If (X,Y)~Normal, then X and Y are independent iff
Cov(X,Y)=0

50. EXPECTED VALUE
If X1 and X2 are independent,

51. CONDITIONAL EXPECTATION AND VARIANCE

52. CONDITIONAL EXPECTATION AND VARIANCE
(EVVE rule)
Proofs available in Casella & Berger (1990), pgs. 154 & 158

53. Example
An insect lays a large number of eggs, each surviving with probability p. Consider a large number of mothers. X: number of survivors in a litter; Y: number of eggs laid
Assume:
Find: expected number of survivors, i.e. E(X)

54. Example - solution =E(Yp) =p E(Y) =p E(E(Y|Λ)) =p E(Λ) =pβ
EX=E(E(X|Y))
=E(Yp)
=p E(Y)
=p E(E(Y|Λ))
=p E(Λ)
=pβ

55. SOME MATHEMATICAL EXPECTATIONS
Population Mean:  = E(X)
Population Variance:
(measure of the deviation from the population mean)
Population Standard Deviation:
Moments:

56. SKEWNESS Measure of lack of symmetry in the pdf.
If the distribution of X is symmetric around its mean ,
3=0  Skewness=0

57. KURTOSIS
Measure of the peakedness of the pdf. Describes the shape of the distribution.
Kurtosis=3  Normal
Kurtosis >3  Leptokurtic
(peaked and fat tails)
Kurtosis<3  Platykurtic
(less peaked and thinner tails)

58. MOMENT GENERATING FUNCTION
The m.g.f. of random variable X is defined as
for t Є (-h,h) for some h>0.

59. Properties of m.g.f. M(0)=E[1]=1
If a r.v. X has m.g.f. M(t), then Y=aX+b has a m.g.f.
M.g.f does not always exists (e.g. Cauchy distribution)

60. Example Suppose that X has the following p.d.f.
Find the m.g.f; expectation and variance.

61. CHARACTERISTIC FUNCTION
The c.h.f. of random variable X is defined as
for all real numbers t.
C.h.f. always exists.

62. Uniqueness
Theorem:
If two r.v.s have mg.f.s that exist and are equal, then they have the same distribution.
If two r,v,s have the same distribution, then they have the same m.g.f. (if they exist)
Similar statements are true for c.h.f.