appm4570-unit-3-pdfs-annotated (1).pdf

appm4570-unit-3-pdfs-annotated (1).pdf - Unit#3 Probability...

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Unit #3: Probability Distributions and Probability Density Functions (Ch 3.4, 3.4, 4.1, 4.2, 4.3) 1
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Learning Objectives At the end of this unit, students should be able to: 1. Distinguish between a continuous and discrete random variable. 2. Distinguish between a random variable and a realization of a random variable. 3. Define a probability mass function (discrete) and a probability density function (continuous) for a rv X. 4. Articulate and justify properties of pmfs and pdfs. 5. Calculate probabilities using pdfs/pmfs. 6. Identify situations for which a Bernoulli, binomial, geometric, or Poisson distribution works as a good model. 7. Calculate the probability that a Bernoulli, Binomial, Negative Binomial, Geometric, or Poisson rv takes on a particular value or set of values. 8. Identify situations for which a uniform, exponential, or beta distribution works as a good model. 9. Calculate the probability that a uniform, exponential, or beta rv falls within a given range. 10.Define the cumulative distribution function (cdf) for a rv. Calculate the cdf for given values of x. 2
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Random Variables Definition : a random variable is a function that maps events to the real numbers. Random variables can be discrete or continuous. Examples: 3 1.) X = # of H in n flips of a fair coin 2.) Y = # of tunes the door opens during class .
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Big Picture: In statistics, we will model populations using random variables, and features/parameters (e.g., mean, variance) of these random variables will tell us about populations. 4
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Random Variables Important Notation: Random variables are usually denoted by uppercase letters near the end of our alphabet (e.g. X, Y ). The value that a random variable takes on is usually denoted by a lowercase letter, such as x, which correspond to the r.v. X. Examples: 5 × = * of flips in 10 tosses E { 0,1 , ... , 10 } X =x rii - T realization of the rv . P(Y=y )
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Probability Density/ Distribution Functions Definition : A Probability distribution function (pdf) is a function that describes the probability distribution of a random variable X. If X is discrete , the pdf provides answers to questions like ________. It is also called a probability mass function (pmf). If X is continuous , then _________ = 0 for all x. Why?! In this case, the distribution function is called a probability density function . In the continuous case, the pdf provides answers to questions like: 6 P ( X=o ) pdf /\ P(X=l ) pmf pdf : ( discrete ) T . cont . P(X=lo ) P(X=x) P(X=x ) P(aeX' . b) , a ,beR .
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Properties of pdfs For f ( x ) to be a legitimate pdf, it must satisfy the following two conditions: 1. 2. (a) For discrete distributions: (b) For continuous distributions: 7 , dist f ( × ) 70 , mass §fcx ) : EPCX =x ) =/ fxf ( x ) d. density
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Discrete Random Variables 8
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Probability Distributions (Discrete) The probability distribution function (pdf) _______ of a discrete r.v. X describes how the total probability is distributed among all the possible range values of the r.v. X . That is: ____________ for each value x in the range of X.
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