Bernoulli random variable
pmf:
p(1) = P(X = 1) = p
p(0) = P(X = 0) = 1 - p = q
E(x) = p
V(x) = p(1 - p) = pq
Binomial Distribution Formulas
n = number of trials
p = probability of success
x = number of successes
pmf:
(; , ) = ( ) (1 )
E(x) = =np
V(x) = 2
Basics of Combinatorics
Theorem
The number of combinations of k objects out of n, without repetition
is
n
= n Ck
k
n!
=
(n k)!k!
Theorem
n
Pk = k!n Ck
n!
=
(n k)!
The number of permutations of k objects out of n, without repetition is
Exercise
8 books a
Continuous Random Variable
Denition
(Continuous random variable). A random variable X is said to be
a continuous random variable if there exists a function f such that
x
FX (x) =
f(t) dt
Denition
Let
x
FX (x) =
f(t) dt
Then, f(t) is called the probability
Conditional Probability
Denition 10. For two events A and B in sample space , where P(B) > 0, the
conditional probability, i.e. the probability that A will occur after B has already
occurred is dened as
P(A|B) =
P(A B)
P(B)
Exercise
A die is rolled once.
Continuous Random Variables
Cumulative Distribution Function
Denition
(Cumulative distribution function). The function
FX = P(X x)
for < x x, is called the cumulative distribution function of the variable X.
Exercise
.
Let X be the result of a die roll. P
Basics of Probability
Terminology
Denition 1 (Experiment). A situation with uncertain results is called an experiment.
Denition 2 (Sample space). The set of all possible outcomes of an experiment
is called the sample space. It is denoted by or S.
Denition
de Moivre-Laplace Limit Theorem
Theorem
(de Moivre-Laplace Limit Theorem). Let Sn be the number of
successes that occur when n independent trials, each with probability of success p,
are performed. Then, for any a < b,
Sn np
< b = (b) (a)
lim P a <
n
np(
Discrete Random Variables
Discrete Random Variables
Denition
(Random variable). A function X : R which maps points from
the sample space to the real line is called a random variable.
Exercise
Three balls are to be randomly selected, without replacement, f
Expectation and Variance
Let X be a continuous random variable. Then, the expectation is
Denition
dened as
xf (x) dx
E[X] =
Exercise
Find the expectation of X is the probability density function is given to be
f(x) =
0<x<1
otherwise
2x ;
0
;
Solution
x f(
Geometric Random Variables
Denition
(Geometric random variable). Suppose that independent trials,
each having probability of success 0 < p < 1, are performed until a success occurs.
If X is the number of trials required, then X is said to have a geometric
Hypergeometric Random Variable
Denition
(Hypergeometric random variable). Suppose that a sample of size
n is to be chosen randomly and without replacement from a population of N , of
which m possess a particular characteristic, and the other N m do not. I
Basic Laws
Law
(Commutative Laws).
AB =BA
AB =BA
Law
(Associative Laws).
(A B) C = A (B C)
(A B) C = A (B C)
Law
(Distributive Laws).
(A B) C = (A C) (B C)
(A B) C = (A C) (B C)
Law
(De Morgans Laws).
A1 An = A1 An
A1 An = A1 An
Proof.
Aa An
/ A An
/
Variance
Denition
(Variance). The variance of a random variable X is dened to be
2
V(x) = E X E[X]
= E X 2 E[X]2
Exercise
Calculate V(X) where X represents the outcome of rolling a fair die.
Solution
E[X] =
6
1
x=1
=
E X2 =
7
2
6
1
x=1
=
6
6
x
x2
91
Population the entire collection of
objects whose properties are to be
analyzed in a particular study.
Sample a subset of the population.
Variable any characteristic of
interest for each object in a
population or a sample.
Observation the set of
measureme
Experiment an activity or process that whose outcome is
subject to uncertainty. That is, experiment lead to random
outcomes.
Sample Space the set of all possible outcomes of an
experiment, denoted by S.
Event any collection (subset) of outcomes of interes
THN’
1. If the percentages of ash content in 12 samples of coal found in close proximity result
a) 287.075
b) 19.319 _ .
c_ 16.943 ‘
d) 4.395 ,
e) 15.708
2. A certain system can experience three different types of defects. Let A (i = 1, 2, 3) denote th
Negative Binomial Random Variable
Denition
(Negative binomial random variable). Suppose that independent
trials, each having probability of success 0 < p < 1, are performed until a total of
r successes are accumulated. If X is the number of trials require
Normal Distribution
Denition
(Normal Random Variable). A random variable X is said to be a
normal random variable if its probability density function is
f(x) =
(x)2
1
e 22
2
where and 2 are parameters.
If
=0
2 = 1
then X is said to be a standard normal r
Uniform Random Variable
Denition
(Uniform Random Variable). A random variable X is said to be a
uniform random variable over the interval (a, b) if its probability density function is
1
; a<x<b
f(x) = ba
0
; otherwise
It is denoted as
X U(a, b)
The cumula
Special Distributions
Bernoulli and Binomial Random Variables
Denition
(Bernoulli random variable). A random variable X is said to be a
Bernoulli random variable if its probability mass function is given by
P(X = 0) = 1 p
P(X = 1) = p
Theorem 15. For a Be
Assumptions for Poisson Distributions for Events Over
a Period of Time
1. The probability that an event occurs in an interval of length h is h + o(h).
2. For a small enough h, the probability that two or more events occur in an
interval of length h is sma
Poisson Random Variables
Denition 25 (Poisson Random Variables). A random variable X that takes on
whole number values is said to be a Poisson random variable with parameter if
for some > 0,
P(X = i) =
e i
i!
where i W.
It is denoted as
X Poi(p)
Theorem 1
Axioms of Probability
Denition
(Probability). The probability of an event E is dened to be a
function which satises the three basic axioms. It is denoted by P (E).
Axiom 1.
0 P(E) 1
Axiom 2.
P() = 1
Axiom 3. For any sequence of mutually exclusive events A
Independent Events
Denition 12 (Two independent events). Two events, A and B, are said to be
independent if
P(A B) = P(A) P(B)
Theorem
P(A|B) = P(A)
if and only if A and B are independent.
Theorem
and B
If A and B are independent, then so are A and B, A a