Chapter 6Continuous Random VariablesA continuous random variable can take any numerical value in someinterval. Assigning probabilities to individual values is not possible.Probabilities can be measured in a given range.For a continuous random variableXwith a numerical value ofinterestxthecumulative distribution function(CDF) is denotedby:with)xX(P)xX(P)x(F<=≤=0)xX(P==For two numerical valuesaandb, witha < b, the probability thatthe outcome is in a range is:)a(F)b(F)aX(P)bX(P)bXa(P)bXa(P−=<−<=≤≤=<<Chapter 6 1
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Theprobability density function(PDF) is given by:for all values ofx.0)x(f>The properties of a probability density function can be illustratedwith a special distribution called theuniform distribution.The uniform distribution over the interval [0, 1] has the PDF:⎪⎩⎪⎨⎧<<=otherwise1x0for01)x(fA graph of the probability density function is below.101a0f(x)xArea = P(X < a) = F(a)Chapter 6 2