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Chapter 4

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University of Guelph

Statistics

STAT 2060

Peter Kim

Winter

Description

Chapter 4- Random Variables and Probability Distributions
Random Variables
Random variable- variable that assumes numerical values associated with the random outcomes of an
experiment, where one (and only one)numerical value is assigned to each sample point
-those that assume countable number (infinite or infinite) of values are called discrete
-those that assume values corresponding to any of the points contained in one or more intervals
(ie. values that are infinite and uncountare called continuous
Probability Distributions for Discrete Random Variables
Requirements for the probability distribution of a discrete random variable, x
●p(x)≥0 for all x
●∑ (x)=1
The mean or expected value of a discrete random variable: μ=E(x)= ∑
The variance of a discrete random variable x is: σ =E [(x- μ) ]=
The standard deviation of a discrete random variable is equal to the square root of the variance (ie.
σ=√ )
Probability Rules for a Discrete Random Variable:
Probabilities for x Chebyshev’s Rule Empirical Rule
-applies to any -applies to any probability distribution
probability distributiothat are mound shaped and symmetric
P(µ − σ < x < µ + σ) ≥ 0 ≈0.68
P(µ − 2σ < x < µ + 2σ) ≥0.75 ≈0.95
P(µ − 3σ < x < µ + 3σ) ≥8/9 ≈1.00
The Binomial Distribution
Binomial experiment consists of:
1. The experiment consists of n identical trials
2. There are two possible outcomes for each trial, ‘success’ or ‘failu(can think in terms of a coin toss with
success being heads and failure being tails)
3.-probability of S remains the same from trial to trial. This probability is denoted by p, and the
probability of F is denoted by q…. P(F) q=1-p
4.each random variable (trial) will be independent
5.binomial random variable x is the number of success in n trials
Binomial Distribution: for discrete random variable is x=0, 1….n is:
P(x)= (
p=probability of success for single trial
q=1-p
n=number of trails
x=number of success in n trials
n-x=number of failures in n trials Mean, Variance, And Standard Deviation for Binomial Random Variable:
-mean µ = np.
-variance σ = npq
-standard deviation σ =√npq
Other Discrete Distributions: Poison and Hypergeometric
Characteristics of a Poison Random Variable:
1. The experiment consists of counting the number of times a certain event occurs during a given unit of
time or in a given area or volume (weight, distance, or any other unit of measurement)
2. The probability that an event occurs in a given unit of time, area, or volume, is the same for all the
units
3. The number of events that occur in one unit of time, area, or volume is independent of the number
that occur in any other mutually exclusive unit
4. The mean (or expected) number of events in each unit is denoted by the Greek letter lambda,
Probability Distribution, Mean, and Variance for a Poison Random Variable:
P(x)= (x-0, 1, 2…)
µ =
σ = e=2.71828…
Characteristics of a Hypergeometric Random Variable:
1. The experiment consists of randomly drawing n elements without replacement from a set of N
elements, r of which are S’s (successes) and (N-r) of which are F’s (failures)
2. The hypergeometric random variable x is the number of S’s in the draw of n elements
Probability Distribution, Mean, and Variance of Hypergeometric Random Variable
( )( )
P(x)= ( ) N=Total number of elements R=number of S’s in the N elements
n=number of elements drawn x=number of S’s drawn in the n elements
The Normal Distribution
Probability Distribution for a Normal Random Variable x
( )
Proba

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