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Today Today: Chapter 5 Reading: –Chapter 5 (not 5.12) –Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62

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Chapter 5 Continuous Random Variables Not all outcomes can be listed (e.g., {w 1, w 2, …,}) as in the case of discrete random variable Some random variables are continuous and take on infinitely many values in an interval E.g., height of an individual

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Continuous Random Variables Axioms of probability must still hold Events are usually expressed in intervals for a continuous random variable

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Example (Continuous Uniform Distribution) Suppose X can take on any value between –1 and 1 Further suppose all intervals in [-1,1] of length a have the same probability of occurring, then X has a uniform distribution on (-1,1) Picture:

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Distribution Function of a Continuous Random Variable The distribution function of a continuous random variable X is defined as, Also called the cumulative distribution function or cdf

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Properties Probability of an interval:

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Example Suppose X~U(-1,1), with cdf F(x)=1/2(x+1) for –1<x<1 Find P(X<0) Find P(-.5<X<.5) Find P(X=0)

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Example Suppose X has cdf, Find P(X<1/2) Find P(.5<X<3)

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Distribution Functions and Densities Suppose that F(x) is the distribution function of a continuous random variable If F(x) is differentiable, then its derivative is: f(x) is called the density function of X

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Distribution Functions and Densities Therefore, That is, the probability of an interval is the area under the density curve

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Example Suppose X~U(0,1), with cdf F(x)=x for –1<x<1 What is the density of X? Find P(X<.33)

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Properties of the Density

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Example (5-16) Suppose X is a random variable and it is claimed that X has density f(x)=30x 2 (1-x) 2 for 0<x<1 Is f(x) a density? If yes, find the c.d.f. of X.

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Example (5-15) Suppose X is a random variable and X has density f(x)=c(1-|x|) for |x|<1 and c is a positive constant Find c? Draw a picture of f(x) Find P(X>1/2)

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Example X has an exponential density: Find F(x)

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Example X has an exponential density: Find the density of Y=X 1/2

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Transformations If Y=g(x) is a one-to-one function with inverse, g -1 (x), the density of Y can be obtained from the density of X as,

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Example X has an exponential density: Find the density of Y=X 1/2

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Example (5-21) Suppose X~U(-1,1) Find the density of Y=|X| Find P(-.5<Y<.75)

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