Stat 48 Lecture May 9thNormal DistributionDr Linda PenasSpring 20071Continuous Distributions (p. 106)DEFN: A random variable is said to be continuousif its values can be put into a 1-1 correspondence to the real numbers.(i.e., measured on a continuous scale)Continuous Prob. DistributionsDEFN: probability density function(pdf): for a continuous random variable X is a curve such that the area under the curvebetween two points a and b is equal to the probability that the random variable X falls between a and b.Continuous Prob. DistributionsCumulative Distribution Function (cdf) of a continuous random variable X is given by( )()( )xF xP Xxf t dt−∞=≤=∫•P(a < X < b) = area under the curve between a and b = F(b) - F(a) abP(a < X < b)•pdf must be nonnegative and the entire area under the curve is 1.•For continuousrandom variables only,()()= ()()P aXbP aXbP aXbP aXb≤≤=<≤≤<=<<Q: Why does that work for continuous rvs?A: Because the area of a single point = 0P(X = a) = 0()( )0aaP aXaf x dx≤≤==∫
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