Continuous probability
distributions
Exponential distribution
Gaussian (normal) distribution
Lognormal distribution
Continuous sample space: deals with measurable
continuous variables changing within a certain range.
Ex: Velocity of a car, amount of alc
Statistical decision theory
Tests of hypotheses and significance
So far.
Gaussian (normal) distribution: Most of observations are
normally distributed.
Standardizing normal variables: calculate the probability of a
normal variable using normal standard t
Discrete probability distribution
functions
Binomial distribution (& Bernoulli distribution)
Multinomial distribution
Hypergeometric distribution
Poisson distribution
Probability distribution functions
Two rules for any probability distribution:
The val
Summarizing statistical data:
frequency distributions, mean value,
mode, median, standard deviation,
variance
Deterministic and stochastic processes
In natural sciences, if the values of parameters for
a problem are known, the exact solution can be
compu
Basic sampling theory
Sampling methods
Sampling distributions
Central limit theory
Sampling theory: studies the relationship between a population
and samples drawn from this population.
Samples randomly selected from a population do not entirely
represe
Geometric distribution
Conditions for a geometric distribution
The geometric distribution is a discrete distribution
dealing with independent events with 2 outcomes
success and failure.
A trial is repeated until a success occurs.
The repeated trials are i
Summary
Discrete distributions for discrete data
Binomial
Multinomial
Hypergeometric
Poisson
Geometric
Pascal
Continuous distributions for continuous data
Exponential
Gaussian (normal) distribution
z
-3 -2 -
+ +2 +3
data
x
Calculate area to find prob
Summary of the last lecture:
Probability of independent events:
Prcfw_E1, E2, ., En=Prcfw_E1 Prcfw_E2. Prcfw_En
1st 2nd
Conditional probability of dependent events:
Prcfw_E1E2= Prcfw_E1 Prcfw_E2|E1
Permutation (order does matter):
Multiset permutation:
n!
Probability theory
Probability
Conditional probability
Permutation
Multiset permutation
Combination
Probability is a measure of the likeliness of an event to
occur: it deals with stochastic processes.
This mathematical branch investigates random variabl
MAT271E Probability and Statistics
Due date, December 8th, 2014
Homework set 9: Answer only 1 question from the problem set.
Q1- (D. A. amaz, Problem 9.13) Car tire wears tests will be carried out for two brands A
and B. For this purpose, samples of 6 car
Statistical estimation theory
Confidence intervals for
Continuous and binomial populations
So far.
We have studied:
Descriptive statistics (how to collect and describe raw data)
Probability (how to find probabilities)
Discrete and continuous probability d
MAT 271 E
Probability and Statistics
Assist. Prof. zge Krkolu-Levitas
[email protected]
Syllabus
Purpose of this course:
To introduce the counting techniques
To introduce the concept of probability
To introduce the basic elements of probability
To make
Statistics
(Statistical Decisions and Hypotheses,
Test of Hypotheses)
STATISTICAL DECISION
Statistical Decisions:
Statistical decisions are decisions made on the basis of observations of a phenomenon
that obeys probabilistic laws that are not completely k
The Uniform Distribution
A random variable X is said to be uniformly distributed in a x b if its density
function is:
f (x) =
1/(b-a)
axb
0
otherwise
The distribution function is given by:
F(x) = P(X x) =
0
(x-a)/(b-a)
1
x<a
ax<b
xb
The Cauchy Distributio
Tree Diagrams:
Calculating probabilities can be hard, sometimes you add them, sometimes you multiply
them, and often it is hard to figure out what to do . tree diagrams to the rescue!
A probability tree diagram shows all the possible events. The first eve
Statistics-Cont.
(Sampling Theory-Cont., Estimation
Theory )
Central Limit Theorem (CLT):
Remind:
The Central Limit Theorem basically says that for non-normal data, the
distribution of the samples means has an approximate normal distribution
with the mean
WEEK 5
Probability-cont.
(Chebyshev Inequality, Other Measures of
Central Tendency and Dispersion,
Skewness and Kurtosis, Some Special
Probability Distributions)
Chebyshev Inequality
Remember: Variance describes how spread apart the values of a random var
Concept of Probability
(Review of Set Theory, Random variables, Axioms,
Theorems, Conditional probability, Bayes Theorem,
Counting Techniques, Tree Diagrams)
Terminology for Probability
Trial: each time you repeat an experiment
Set: a well-defined colle
Statistics-Cont.
(Confidence Level and Intervals)
ESTIMATION THEORY
Remind:
There are two types of estimates for parameters:
Point estimate (given by a single number)
Its is difficult to find and its reliability?!
Interval estimate (by two numbers betwe
Probability-cont.
(Continuous Probability Distributions,
Joint Distributions, Expectation, Variance
and Std. Dev. of a Random Variable,
Standartization, Covariance)
Continuous Probability Distributions
Remember: a continuous random variable is the one whi
Statistics
(Sampling Theory)
Some Definitions of Sampling
Theory
Population (Universe): A group of individuals under study is called a
population or universe. Population size is denoted by N. If N is finite then it
is called a finite population. If N is
WORKSHEET-5
Course Title: Mat101E
Content: Integration
1. Evaluate the following integrals
Z
t t+ t
(a)
dt,
t2
Z
9r2
(b)
dr,
1 r3
Z
sin(2t + 1)
(c)
dt,
cos2 (2t + 1)
18 tan2 x sec2 x
dx,
(d)
(2 + tan3 x)
Z
cos
d,
(e)
sin2
Z
1
1
1
(f)
cos d,
2 sin
Z
(
WORKSHEET-3
Course: MAT101E
Subject: Derivatives
1. Using the definition, find the derivatives of the following functions. Then evaluate the
derivatives at the specified points.
(a) f (x) = (x 1)2 + 1; f 0 (1), f 0 (0), f 0 (2).
1 ; f 0 (1), f 0 (2).
(b)
WORKSHEET 8
Course: Mat101E
Content: Applications of Integrals
1. A plane slices a ball of radius a into two pieces. If the plane passes b units away from the centre of
the ball (where b < a), find the volume of the smaller piece, using the method of slic
WORKSHEET 4
Course: Mat101E
Content: Applications of Derivatives
1. Find the absolute extreme values of the following functions on the given interval. Then
graph the function. Identify points on the graph where absolute extreme occur and include
their coo