ch-4-3502 - CHAPTER 4 Continuous Random Variables and...

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CHAPTER 4 Continuous Random Variables and Probability Distributions z Basic definitions and properties of continuous random variables Continuous random variable: A random variable is continuous if its set of possible values is an entire interval of numbers. Probability distribution for continuous variables: It is possible to construct a prob- ability histogram (same as relative frequency histogram) for continuous variable. But by measuring the variable more and more finely, the resulting histogram approaches to a smooth curve. It is obvious that the total area under this curve is 1, also probability that the variable be between two points is the area under the curve between two points . It means probabil- ity distribution or probability density function (pdf) for a continuous random variable X is a function f ( x ), such that P ( a 6 X 6 b ) = Z b a f ( x ) dx f ( x ) is a pdf if satisfies the following two conditions: f ( x ) > 0 for all x R -∞ f ( x ) dx = 1 = area under the entire curve of f ( x ) ——————————————————————————— Example: Suppose that X has following density function f ( x ) = ( 0 . 5 x 0 6 x 6 2 0 otherwise. Calculate a. P ( X 6 1) b. P (0 . 5 6 X 6 1 . 5) c. P (1 . 5 6 X ) —————————————————————————– Note: For a continuous random variable P ( X = C ) = 0, then P ( a 6 X 6 b ) = P ( a < X 6 b ) = P ( a 6 X < b ) = P ( a < X < b ) ——————————————————————————– Example: Consider the following function f ( x ) = ( 0 . 15 e - 0 . 15( x - 0 . 5) x > 0 . 5 0 otherwise. 1
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a. Verify that f ( x ) is a pdf b. P ( X 6 5) ——————————————————————————— z Uniform Distribution A continuous random variable X has uniform distribution on interval [ a,b ] if the pdf of X is f ( x ; a,b ) = ( 1 b - a a 6 X 6 b 0 otherwise. z Cumulative Distribution Function The cumulative distribution function (pdf) for a continuous random variable is F ( x ) = P ( X 6 x ) = Z x -∞ f ( y ) dy x F ( x ) is the area under the density curve to the left of x . ———————————————————- Example: Find cdf for uniform distribution on [ a,b ] and then graph it. ———————————————————— Same as discrete random variable, the probabilities of intervals can be com- puted from F ( x ) as P ( X > 0) = 1 - F ( a ) , P ( a 6 X 6 b ) = F ( b ) - F ( a ) —————————————————————————– Example: Solve example 1 by using cdf. —————————————————————————– If
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ch-4-3502 - CHAPTER 4 Continuous Random Variables and...

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