STAT 230MIDTERM 1

QUANTITATIVE SCIENCES COURSE UNIONqscu.org

IntroductionsName, Major, Role in QSCU

WHAT IS QSCU?

How do I get involved?Weekly Meetings:

Tuesdays, 3:30 - 5:00 UNC Boardroom

OPEN TO EVERYONE! (Snacks are sometimes provided)

How do I stay in the loop?

Facebook.com/ubcoqscuQSCU.org

● What is Statistics?● Probability● Counting and Bayes Theorem● Random Variables and DisCrete Distributions● Special Discrete Probability Distributions● Expected Values and variances

Table of content

WHAT IS statistics?

Types of qualitative data● Categorical/Nominal

○ Can take on one of a limited number of possible values○ ex) Nationality: Irish, English, Canadian, American,

French, etc.● Ordinal

○ Exist on an ordinal scale○ ex) Rating from 1 to 10

● Binary○ Data which can only take on two possible values○ ex) yes or no

Types of qualitative data● Continuous

○ Data arises when all values are possible inside some interval on the real line

○ ex) distance between cities● Discrete

○ Data arises when the possible items are countable

○ ex) number of flips needed to get 10 heads with a flip of a coin

Types of qualitative data

Population & Sample● Difference between population and sample● Population

○ It is an all-encompassing group of interest○ Usually unobservable in its entirety for one reason

or another(most often the cost of measurement)○ ex) UBC Okanagan Students

● Sample○ Number of observations○ ex) people sitting in this classroom is a sample of

UBC students

Percentile & Quartile & Interquartile Range● Percentile

○ Value that has 100p% of ordered data falling below it○ Value that has 100(1-p)% of the ordered data falling

above it● Quartile (Q)

○ 25th percentile = first quartile (Q!)○ 75th percentile = third quartile (Q3)

● Interquartile Range (IQR)○ Size of the gap between the first and third quartile○ ‘Distance’ over which the ‘middle half’ of the data is

spread○ IQR = Q3 - Q1

probability

Axioms of Probability

- Events cannot happen at the same time

- Check if disjointa. A and B = 0

- ex) a dice cannot be 2 and 3 at the same time

b. P(A∩B) =0c. Check the formula

- P(AUB) = P(A)+P(B)- Check LHS = RHS

Disjoint/Mutually Exclusive

Disjoint/Mutually Exclusive QuestionsQ1. Tossing a coin : Heads and Tails

Q2. Cards: Kings and Hearts

Q3. Turning left and Turning right

Q4. Turning left and Scratching head

Q5. Cards: Kings and Aces

- The outcome of A does not affect B- ex) rolling a dice, then rolling again- Check formula

a. P(A and B) = P(A)P(B)b. Check LHS = RHS

Independent events

- Recall:- Disjoint

- P(AUB) = P(A)+P(B)- Because P(A∩B) =0

OR Rule

Generalized OR Rule

- Second event is not independent from the first event

Conditional Probability

AND Rule

Generalized AND Rule

Complement Rule

A die is rolled 30 times, what is the probability of getting 3 at least once?

- Let T be the event that i(number of dice) dice getting 3

- Let E be the event getting 3 at least once

Counting and bayes theorem

Fundamental principle of counting

- Factorial- The number of ways of arranging n DISTINCT objects

in a line is- n! = n*(n-1*)(n-2)*...*3*2*1

- Permutation- Each of n! Arrangement of n distinct objects is called a

permutation of the n objects.- ORDER DOES MATTER

Factorial & Permutation

Permutation & Factorial

Combination- Order does not matter

Conditional Probability & Bayes Theorem Equation- The probability that an event E occurs given that an event

E has already occured

Partition Theorem- All events F must be mutually exclusive and cover entire

sample space

Partition Theorem

Partition Theorem

Bayes Theorem

Random variables & discrete distributions

- variable that follows a probability distribution - -maps the sample space to the real numbers

Random variable

Probability mass function

Probability mass function

Probability mass function

Cumulative distribution function

- Plot of cumulative distribution function looks like a staircase all the way up to one

Cumulative distribution function

Cumulative distribution function1. Using pmf table from slide 48, create cdf table.2. Find :

a. F(4)b. F(8)c. F(-2)d. F(2.5)

propositions

propositions1. Using the same example from slide 48 and slide 51,

Find :

a. P(2≤x≤4)b. P(4≤x≤6)c. P(2<x≤5)

sPecial Discrete Probability distributions

- Random variable whose possible values are 0 and 1- ex) binary, yes or no

- Single trial

Bernoulli trial

- Random variable - X ~ Bernoulli(p)

- P is parameter = probability

Bernoulli distribution

1. Multiple Bernoulli trials2. 4 Requirements:

a. The experiment consists of a sequence of n trials, where n is fixed in advance.

b. There are only two possible outcomes for each trial (success or failure)

c. The trials are independent. (The outcome one trial does not influence the outcome of another.)

d. The probability of “success” on each trial does not change (constant p).

binomial distribution

- Random variable - X ~ Binomial (n,p)

- Parameter n = number of trials- p = probability

binomial distribution

binomial distribution

binomial distribution

- Counts over a period of time or space (time/space interval)- ex) number of Stats student entering the Math and Science Center

an hour

Poisson distribution

- Lambdaa. MAKE SURE THE AVERAGE RATE (LAMBDA) IS

CONSISTENT WITH TIME INTERVAL- 3 Requirements

a. Events are independent of each other.b. Lambda is constant.c. Two events cannot occur at the same time.

Poisson distribution

Poisson distribution

Expected values & variances

- Expected value of X- Average or Mean value of X

Expected value (EV)

- The expected value can be interpreted as a long run average, a weighted average of all possible values, or as a center of mass of the distribution.

Interpretation of Expected value (EV)

- Using the same example from slide 48 and slide 51,

Find:

E[X] for discrete random variable

Expected value (EV)

Properties of expected values

Variance & standard Deviation

Variance & standard Deviation

Variance & standard Deviation

Properties of variance

Mean and variance for special distribution

Good luck!!