Transcript Lecture 7

Statistics 270 - Lecture 7
• Last day: Completed Chapter 2
• Today: Discrete probability distributions
• Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110
Example (Chapter 2 - 78)
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A system of components is connected as in the following diagram.
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Components 1 and 2 are connected in parallel so the subsystem works if and only if
either 1 or 2 works
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Components 3 and 4 are connected in series, so the subsystem works iff and only iff
both 3 and 4 work
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Assume that the system works iff either the first or second subsystem works
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Assume that the components work independently of each other and P(Component
works)=0.9
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Find P(system works)
Example (Chapter 2 - 78)
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Example – Let’s Make a Deal:
• A contestant is given a choice of three doors of which one
contained a prize such as a Car
• The other two doors contained gag gifts like a chicken or a donkey
• After the contestant choses an initial door, the host of the show
reveals an empty door among the two unchosen doors, and asks
the contestant if they would like to switch to the other unchosen
door
• What should the contestant do?
Example
• Roll two dice
• Events:
• A1={first die response is odd}
• A2={second die response is odd}
• A3={Sum of dice is odd}
• Are the events mutually independent?
Another Example
• N people go to a restaurant and check their coats
• The coats are given back randomly
• What is the probability that no one receives their own coat
Chapter 3 – Discrete Random Variables
• Recall – an experiment is a process where the outcome is uncertain
• The experiment can take on a variety of outcomes
• Each outcome can be associated with a number by specifying a rule
of association….a random variable is such a rule
• Random Variable: For a given sample space, S, a random
variable is any rule that associates a number with each outcome in
S
• Can be viewed as a map from the sample space to the real line
• Will consider two types:
• Discrete random variables
• Continuous random variables
Discrete versus Continuous
• Discrete random variables have either a finite number of values or
infinitely many values that can be ordered in a sequence
• Continuous random variables take on all values in some interval(s)
Examples
• Discrete or continuous
• Number of people arriving in a supermarket
• Hair color of randomly selected people
• Weight lost from a diet program
• Random number between 0 and 4
Discrete Random Variables
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Describe chances of observing values for a discrete random variable by
probability distribution or probability mass function
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Probability distribution of a discrete random variable, X, is the list of distinct
numerical outcomes and associated probabilities
Value of X
x1
x2
…
Probability p(xi) p(x1) p(x2) …
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P(X=xi)=p(xi)
xk
p(xk)
Example
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Flip a coin
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Get responses heads or tails
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S={H,T}
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X(H)=1
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Random variable X takes on value 1 for heads and 0 for tails
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A rv that takes on two values, 0 and 1, is called a Bernoulli rv
X(T)=0
Properties
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p ( xi ) 
k
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 p( x ) 
i 1
i
for each value xi of X
Example
• Consider a baseball player with a 300 batting average (i.e., gets hit 30% of
the time)
• Let X be the number of at bats until the batter gets a hit
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Describe the probability distribution for X
• Can display distribution using a probability histogram
• X-axis represents outcomes
• Y-axis is the probability of each outcome
• Use rectangles, centered at each value of X, to display probabilities
Example
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Probability distribution for number people in a randomly selected household
X=# people
p(xi)
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0.25 0.32 0.17 0.15 0.07 0.03
Draw the probability histogram