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
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
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
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
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
/
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
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
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