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Lecture 11

Lecture 11.docx

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McMaster University
Jeff Racine

Lecture 11 Continuous Probability Density Functions Continuous random variables Unlike their discrete counterparts, continuous random variables are ‘uncountable’ and can assume any value on the real number line Rather than looking at p(X =x) (i.e. the probability of any specific occurrence), we instead look at p( a < X < b), i.e., the probability that a continuous random variable lies in an interval The probability density function Commonly shows probabilities with ranges of values along the continuum of possible values that the random variables might take on Let some smooth curve represent how the probability of a continuous random variable X is distributed. If the smooth curve can be represented by a formula f(X), then the function f(X) is called the probability density function Areas will be seen to play an important role in determining probabilities Since there exist an infinite number of points in an interval, we cannot assign a non-zero probability to each point and still have their probability sum to one, therefore, the probability of any specific occurrence of a continuous random variable is defined to be zero The total area under the probability density function (just as under the relative frequency histogram) must therefore equal 1 We shall study two common distributions The uniform probability distribution The normal probability distribution Basic Rules for Probability Density Functions If a smooth curve is to represent a probability density function, then the following two requirements must be met: The total area between the curve and the horizontal axis must be unity, i.e., ∞ ∫ f (x)dx=1 −∞ The curve must never fail below the horizontal axis, i.e., f(x)
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