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Stat 362: Chapter 3 Worksheet

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Show all of your work / code. No credit will be given if there is no work / code.

For each of the below questions, report your code and results.

1. In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of

sample points in the sample space S for the number of possible orders at the conclusion of

the contest for

(a) all 8 finalists;

(b) the first 3 positions.

2. How many ways are there to select 3 candidates from 8 equally qualified recent graduates for

openings in an accounting firm?

3. A president and a treasurer are to be chosen from a student club consisting of 50 people. How

many different choices of officers are possible if

(a) there are no restrictions?

(b) Abe will serve only if he is president?

4. In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail

to complete the test run without a blowout. Of the next 15 trucks tested, find the probability

that (Hint: Binomial Distribution)

(a) less than 4 have blowouts;

(b) from 3 to 6 (inclusive) have blowouts.

5. An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The

distribution of the number of cars per year that will experience the catastrophe is a Poisson

random variable with λ = 5. What is the probability that greater than 1 car per year will

experience a catastrophe?

6. Suppose P(T > k) = 0.05, where T comes from the t-distribution with 10 degrees of freedom.

Find k. Hint: Use the function qt()

7. Generate a sample of size 30 from the Gamma Distribution with parameters α = 3 and β = 5

(scale parameter). We want to be able to replicate this sample, so use a seed of 15. Report

the sample mean and sample variance along with your code.

2

Probability and Distributions

Chapter 3

Stat 362

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Outline

1 Random Sampling

2 Probability Calculations and Combinatorics

3 Discrete Distributions

4 Continuous Distributions

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Random Sampling

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Types of Sampling

Why random sample?

Types of random sampling:

Simple Random Sample (SRS)

Stratified Sampling (STSRS)

Cluster Sampling

Systematic Sampling

Multistage Sampling

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Simple Random Sampling

Simple Random Sample:

sample(x, size, replace = FALSE, prob = NULL)

x: vector of numbers to select from

size: how many numbers you want to get

replace: sample with replacement (TRUE), sample without

replacement (FALSE)

prob: vector of probabilities. Default is uniform (each value has the

same probability of being selected)

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How to generate a replicable series of random variables

Set a random number seed. Allows for replication to generate the

exact same dataset each time you run your code. Must be run right

before the random sample generation.

set.seed()

Example

set.seed(16)

round(rnorm(n = 10, mean = 30, sd = 5),2)

## round() rounds values to a certain

## number of decimal places

Output:

32.38 29.37 35.48 22.78 35.74

27.66 24.97 30.32 35.12 32.87

See Additional R Code: Random Sampling

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Probability Calculations and Combinatorics

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Permutations, Combinations

Permutations can be written as a division of factorials.

factorial() returns the value of a factorial.

Example: factorial(4) = 4(3)(2)(1) = 24.

prod() gives the product of the values inside the function.

Example: prod(c(3,4)) = 3(4) = 12.

Combinations can be written with factorials, products, or the

binomial coefficient.

choose(n,k) returns the value of

n

k

See Additional R Code: Probability calculation and combinatorics

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Discrete Distributions

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Binomial Distribution

Let X represent the number of successes in n Bernoulli trials. Then

X ∼ Bin(n, p).

PDF:

n x

P(X = x) =

p (1 − p)n−x , x = 0, 1, 2, . . . , n.

x

Expected Value:

E(X ) = np

Variance:

V(X ) = σ 2 = np(1 − p)

Standard Deviation:

SD(X ) = σ =

p

np(1 − p)

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Binomial Distribution: R Functions

PDF: dbinom(x, size = , prob = )

CDF: pbinom(x, size = , prob = )

Quantile: qbinom(q, size = , prob = )

The smallest value x such that F (x) ≥ p,

where F is the distribution function.

Generate Random Sample: rbinom(n, size = , prob = )

Example

Suppose X ∼ Bin(20, 0.3)

P(X = 5): dbinom(5, size = 20, prob = 0.3)

P(X ≤ 5): pbinom(5, size = 20, prob = 0.3, lower.tail= TRUE)

50th percentile (median): qbinom(0.5, 20, 0.3, lower.tail = TRUE)

Generate a random sample of size 10:

rbinom(n = 10, size = 20, p = 0.3)

Note: lower.tail = TRUE is optional in the pbinom and qbinom functions

because it is the default setting.

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Poisson Distribution

Let X represent the number of outcomes occurring in a given time

interval or specified region. Let λ represent the average number of

outcomes per unit time, distance, area, or volume. Then X ∼ Pois(λ).

PDF:

P(X = x) =

e −λ λx

, x = 0, 1, 2, . . . .

x!

Expected Value:

E(X ) = λ

Variance:

V(X ) = σ 2 = λ

Standard Deviation:

√

SD(X ) = σ =

λ

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Poisson Distribution: R Functions

PDF: dpois(x,λ)

CDF: ppois(x,λ)

Quantile: qpois(q,λ)

It is the smallest integer x such that P(X ≤ x) ≥ p.

Generate Random Sample: rpois(n,λ)

Example

Suppose X ∼ Pois(λ = 2.5).

P(X = 1): dpois(1,2.5)

P(X ≤ 5): ppois(5,2.5, lower.tail = TRUE)

50% percentile (median): qpois(0.5,2.5, lower.tail = TRUE)

Generate a random sample of size 10: rpois(10,2.5)

Note: lower.tail = TRUE is optional because it is the default setting.

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Continuous Distributions

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Normal (Gaussian) Distribution

Let X ∼ N(µ, σ 2 ).

PDF:

f (x) =

2

1 − (X −µ)

2σ 2 ,

−∞

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