09_20 - 0 x 4, f ( x ) = 0, otherwise. a) What must the...

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STAT 400 Examples for 09/20/2011 Fall 2011 Continuous Random Variables. The probabilities associated with a continuous random variable X are determined by the probability density function of the random variable. The function, denoted ƒ ( x ), must satisfy the following properties: 1. ƒ ( x ) 0 for all x . 2. The total area under the entire curve of ƒ ( x ) is equal to 1.00. Then the probability that X will be between two numbers a and b is equal to the area under ƒ ( x ) between a and b. For any point c, P(X = c) = 0. Therefore, P(a X b) = P(a X < b) = P(a < X b) = P(a < X < b). Expected value (mean, average): - = x x x d f ) ( X μ . Variance: σ X 2 = ( ) [ ] ( ) - - = - dx x x f ) ( X E 2 X 2 X μ μ . σ X 2 = ( ) ( ) [ ] ( ) 2 X 2 2 2 μ ) ( X E X E - = - - dx x x f . Moment Generating Function: M X ( t ) = E ( e t X ) = ( ) - dx x f x t e .
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1. Let X be a continuous random variable with the probability density function f ( x ) = k x ,
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Unformatted text preview: 0 x 4, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? b) Find the cumulative distribution function of X, F X ( x ) = P ( X x ). c) Find the probability P ( 1 X 2 ). d) Find the median of the distribution of X. That is, find m such that P ( X m ) = P ( X m ) = 1 / 2 . e) Find the 30th percentile of the distribution of X. That is, find a such that P ( X a ) = 0.30. f) Find X = E ( X ). g) Find X = SD ( X ). 2. Let X be a continuous random variable with the cumulative distribution function F ( x ) = 0, x &lt; 0, F ( x ) = 8 3 x , 0 x 2, F ( x ) = 1 2 1 x , x &gt; 2. a) Find the probability density function f ( x ). b) Find the probability P ( 1 X 4 ). c) Find X = E ( X ). d) Find X = SD ( X )....
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09_20 - 0 x 4, f ( x ) = 0, otherwise. a) What must the...

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