Statistics 312 14 Probability Density Functions 1 Continuous random variables are variables for which any value within a range of values can occur These distributions differ from discrete variables in the following ways 1 Any value within the range of the variable can occur rather than just specific values 2 The probability of occurrence of a specific value X is zero 3 Probabilities can be obtained by cumulating an area under a curve In order to find the probability within an interval from a to b we need to find the probability density function f X for the variable X In general for a continuous variable the probability of falling between a and b is b P a X b a f x dx The cumulative distribution function gives the area probability of being less than a given value of x b The mean or expected value and variance for a continuous variable Example Uniform distribution Uniform pdf f x a b 1 b a a x b 0 otherwise Example Suppose A 2 and B 12 Then a b 2 b a 2 2 12 Statistics 312 14 Probability Density Functions 1 2 x 12 f x 2 12 10 0 otherwise and 1 x7 7 3 4 P 3 x 7 3 dx 3 10 10 10 10 10 7 2 Statistics 312 14 Probability Density Functions b cdf P X b F b 1 dx 10 0 b 2 10 1 x 2 2 x 12 x 12 P 3 x 7 F 7 F 3 5 10 1 10 4 10 3 Statistics 312 E x Note 2 14 Probability Density Functions 12 x 2 12 144 4 140 x dx 2 xf x dx 7 10 20 20 20 20 2 a b 2 12 7 2 2 x m 1728 8 30 2 12 f x dx x 7 10 2 dx x 12 2 2 14 x 49 10 x 3 14 x 2 49 x 12 2 dx 10 30 20 14 144 4 49 12 2 57 33 98 49 8 33 20 10 b a 12 2 Note 2 2 2 12 2 12 100 8 33 12 Read pp 180 183 Prob 5 2 5 3 5 4 4
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