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Random Variables and Probability
Distributions: Lecture IV
I. Conditional Probability and Independence
A. In order to define the concept of a conditional probability it is necessary to discuss joint probabilities and marginal probabilities.
- A joint probability is the probability of two random events. For example, consider drawing two cards from the deck of cards. There are 52x51=2,652 different combinations of the first two cards from the deck.
- The marginal probability is overall probability of a single event or the probability of drawing a given card.
- The conditional probability of an event is the probability of that event given that some other event has occurred. a) In the textbook, what is the probability of the die being a one if you know that the face number is odd? (1/3). b) However, note that if you know that the role of the die is a one, that the probability of the role being odd is 1. B. Axioms of Conditional Probability:
1. P A B 0 for any event A.
2. P A B 1 for any event A B.
3. If Ai B , i 1, 2,are mutually exclusive, then
P A 1 A 2 P A B 1 P A B 2
4. If B H B , G and P G 0 then
P H B P H
P G B P G
C. Theorem 2.4.1: ^ ^
P A B P A^ B
P B
^ for any pair of events A and B such
that P B ^ ^0.
D. Theorem 2.4.2 (Bayes Theorem): Let Events A 1 (^) , A 2 , An be mutually exclusive
such that P A 1^ ^ A 2^ ^ An ^1 and P A i^ ^0 for each i. Let E be an
arbitrary event such that P E 0. Then
Professor Charles B. Moss Fall 2007
(^) n
j j j
i i i PE A P A
PE A P A
P A E
1
- Another manifestation of this theorem is from the joint distribution function: P ( E , Ai ) P ( E Ai ) P ( E | Ai ) P ( Ai )
- The bottom equality reduces the marginal probability of event E
n i
P E P E Ai P Ai 1
- This yields a friendlier version of Bayes theorem based on the ratio between the joint and marginal distribution function:
P E
P E A
P Ai E i
E. Statistical independence is when the probability of one random variable is independent of the probability of another random variable.
- Definition 2.4.1: Events A and B are said to be independent if P A (^) P A B .
- Definition 2.4.2: Events A , B , and C are said to be mutually independent if the following equalities hold: a) P A ^^ ^ B^ ^ P A P B ^ ^ ^ b) P A C (^) P A P C c) P B C P B P C d) P A B C (^) P A P B P C
II. Basic Concept of Random Variables: Casella and Berger section 1.
A. Definition 1.4.1: A random variable is a function from a sample space S into the real numbers. B. In this way a random variable is an abstraction. We assumed that there was a random variable defined on some sample space like flipping a coin. The flipping of the coin is an outcome in an abstract space S s 1 (^) , s 2 , sn
We then define a numeric value to this set of random variables
Professor Charles B. Moss Fall 2007
(^) j
i j i j PY y
PX x Y y P X x Y y
- Definition 3.2.3. Discrete random variables are said to be independent if the event (^) X xi , and the event (^) Y yj are independent for all i j ,. That is to say, P X x Yi , y (^) j P X xi (^) P Y yj .
Table 2. Conditional Probability P [ X , Y =2] P [ Y =2] P [ X | Y =2] P [ X ] 0 0.0581 0.3456 0.1681 0. 1 0.1245 0.3456 0.3602 0. 2 0.1067 0.3456 0.3087 0.308 7 3 0.0457 0.3456 0.1323 0. 4 0.0098 0.3456 0.0284 0. 6 0.0008 0.3456 0.0024 0.
Table 3. Uncorrelated Discrete Normal Distribution
X
Y (^) Marginal 0 1 2 3 4 5 Probability 0 0.005 0.018 0.028 0.005 0.003 0.000 0. 1 0.023 0.055 0.088 0 .070 0.010 0.000 0. 2 0.025 0.068 0.148 0.105 0.028 0.003 0. 3 0.010 0.063 0.100 0.045 0.010 0.003 0. 4 0.003 0.030 0.033 0.015 0.000 0.003 0. 5 0.000 0.000 0.008 0.003 0.000 0.000 0. Marginal Probability 0.07 0.23 0.40 0.24 0.05 0.
Table 4. Conditional Probability X P [ X | Y =2] P [ Y =2] P [ X | Y =2] P [ X ] 0 0.0275 0.4025 0.0683 0. 1 0.0875 0.4025 0.2174 0. 2 0.1475 0.4025 0.3665 0. 3 0.1000 0.4025 0.2484 0. 4 0.0325 0.4025 0.0807 0. 5 0.0075 0.4025 0.0186 0.
Professor Charles B. Moss Fall 2007
Table 5. Correlated Discrete Normal Distribution
X
Y (^) Marginal 0 1 2 3 4 5 Probability 0 0.068 0.025 0.000 0.000 0.000 0.000 0. 1 0.028 0.115 0.078 0.003 0.000 0.000 0. 2 0.000 0.065 0.200 0.060 0.000 0.000 0. 3 0.000 0.000 0.088 0.143 0.020 0. 000 0. 4 0.000 0.000 0.003 0.033 0.043 0.008 0. 5 0.000 0.000 0.000 0.003 0.010 0.013 0. Marginal Probability 0.095 0.205 0.368 0.240 0.073 0.
Table 6. Conditional Probabilities X P [ X | Y =2] P [ Y =2] P [ X | Y =2] P [ X ] 0 0.0000 0.4025 0.0000 0.0 925 1 0.0775 0.4025 0.1925 0. 2 0.2000 0.4025 0.4969 0. 3 0.0875 0.4025 0.2174 0. 4 0.0025 0.4025 0.0062 0. 5 0.0000 0.4025 0.0000 0.
- Theorem 3.2.1. Discrete Random variables X and Y with the probability distribution given in Table 3.1 are independent if and only if every row is proportional to any other row, or, equivalently, every column is proportional to any other column. E. Multivariate Random Variables
- Definition 3.2.4. A T-variate random variable is a variable that takes a countable number of points on the T-dimensional Euclidean space with certain probabilities.
III. Univariate Continuous Random Variables
A. Definition 3.3.1. If there is a nonnegative function f (^) x defined over the whole line such that 2 1
1 2 (^ )
x Px X x x f xdx
for any x 1 , x 2 satisfying x 1 (^) x 2 , then X is a continuous random variable and
Professor Charles B. Moss Fall 2007
(^)
^
0 otherwise
(^1) given 0 x , 0, 0 ,
x x e f x
C. Normal Distribution
forall- x 2 2
2 2
x f x Exp
D. Beta Distribution
(^)
^
0 otherwise
1 for 0 x 1, 0, 0 ,
(^11) x ^ x^ f x
