REVISION SESSIONAL TEST
description
Transcript of REVISION SESSIONAL TEST
REVISIONSESSIONAL TEST
Variables Qualitative variables take on
values that are names or labels. The colour of a ball (e.g., red, green, blue)
Quantitative variables are numeric. They represent a measurable quantity. For example, population of a city
Discrete vs Continuous Variables
Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.
Basic Concepts Consider following set of values:
12, 15, 21, 27, 20, 21 Mean Mode Median Variance Standard Deviation Range
Basic Concepts Consider following set of values:
12, 15, 21, 27, 20, 21 Stem-and-Leaf Plot Box-and-Whisker Plot Frequency Distribution Table Histogram
Trial A single performance of an
experiment is called a Trial. The result of a trial is called an
Outcome or a Sample Point. The set of all possible outcomes
of an experiment is called a Sample Space.
Subsets of a Sample Space are called Events.
Examples – Sample Space The set of integers between 1 and 50 and
divisible by 8 S = { x! x2 + 4x – 5 = 0 } Set of outcomes when a coin is tossed
until a tail or three heads appear Set of sampling items randomly until one
defective item is observed.
Definitions An event is a subset of a Sample Space. The complement of an event A with
respect to S is the subset of all elements of S that are not in A.
The intersection of two events A and B is the event containing all elements that are common to A and B.
The union of two events A and B is the event containing all the elements that belong to A or B or both.
Mutually Exclusive Events Two events A and B or
Mutually Exclusive or Disjoint, if A and B have no elements in common.
ProbabilityIf the Sample Space S of an
experiment consists of finitely many outcomes (points) that are equally likely, then the probability P(A) of an event A is
P(A) = Number of Outcomes (points) in A Number of Outcomes (points) in
S
Permutation A permutation is an arrangement of all or
part of a set of objects. Number of permutations of n objects is n! Number of permutations of n distinct
objects taken r at a time is nPr = n!
(n – r)! Number of permutations of n objects
arranged is a circle is (n-1)!
Permutations The number of distinct
permutations of n things of which n1 are of one kind, n2 of a second kind, …, nk of kth kind is
n! n1! n2! n3! … nk!
Problem How many permutations of 3
different digits are there, chosen from the ten digits, 0 to 9 inclusive?
A84 B 120
C504D 720
Problem In how many ways can a
Committee of 5 can be chosen from 10 people?
A252 B 2,002
C30,240 D 100,000
Independent Probability If two
events, A and B are independent then the Joint Probability is
P(A and B) = P (A Π B) = P(A) P(B)
For example, if two coins are flipped the chance of both being heads is
1/2 x 1/2 = 1/4
Mutually Exclusive If either event A or event B or both events
occur on a single performance of an experiment this is called the union of the events A and B denoted as P (A U B).
If two events are Mutually Exclusive then the probability of either occurring isP(A or B) = P (A U B) = P(A) + P(B)
For example, the chance of rolling a 1 or 2 on a six-sided die is 1/6 + 1/6 = 2/3
Conditional Probability Conditional Probability is the probability
of some event A, given the occurrence of some other event B.
Conditional probability is written as P(A І B), and is read "the probability of A, given B". It is defined by
P(A І B) = P (A Π B) P(B)
Conditional Probability
Consider the experiment of rolling a dice. Let A be the event of getting an odd number, B is the event getting at least 5. Find the Conditional Probability P(A І B).
Independent EventsTwo events, A and B,
are independent if the fact that A occurs does not affect the probability of B occurring.
P(A and B) = P(A) · P(B)
Complementation Rule
For an event A and its complement A’ in a Sample Space S, is
P(A’) = 1 – P(A)
Example - Complementation Rule
5 coins are tossed. What is the probability that:
a. At least one head turns upb. No head turns up
ProblemList following of vowel letters
taken 2 at a time:a. All Permutationsb. All Combinations without
repetitionsc. All Combinations with
repetitions
Probability Mass Function
A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.
Cumulative Distribution Function
x P(x≤A)
1 P(x≤1)=1/6
2 P(x≤2)=2/6
3 P(x≤3)=3/6
4 P(x≤4)=4/6
5 P(x≤5)=5/6
6 P(x≤6)=6/6
Probability Density Function
A probability density function (pdf) describes the relative likelihood for the random variable to take on a given value.
Discrete Distribution: Formulas
P(a < X ≤ b) = F(b) – F(a) = b xj a
Pj
j
1 Pj (Sum of all Probabilities)
Continuous Distribution: Formulas
P(a < X ≤ b) = F(b) – F(a) = b
a
vf )(
)(vf dv = 1 (Sum of all Probabilities)
dv
Uniform Distribution
f(x) = 1/b-a a ≤ x ≤ b 0 Otherwise
F(x) = x-a/b-a a ≤ x ≤ b 0 Otherwise
Review Question
Two dice are rolled and the sum of the face values is six? What is the probability that at least one of the dice came up a 3?a. 1/5b. 2/3c. 1/2d. 5/6e. 1.0
Review Question
Two dice are rolled and the sum of the face values is six. What is the probability that at least one of the dice came up a 3?
a. 1/5b. 2/3c. 1/2d. 5/6e. 1.0
How can you get a 6 on two dice? 1-5, 5-1, 2-4, 4-2, 3-3One of these five has a 3. 1/5