Probability Distributions and Density Functions PDF

(4.1) Probability Distributions and Probability Density Functions Definition: A continuous random variable X is a random variable with an interval of real numbers (finite or infinite) for its range. Definition: For a continuous random variable X , a probability density function (PDF) is a function...

(4.1) Probability Distributions and Probability Density Functions Definition: A continuous random variable X is a random variable with an interval of real numbers (finite or infinite) for its range. Definition: For a continuous random variable X , a probability density function (PDF) is a function such that 1. f ( x ) ≥ 0 ∞ 2. ∫ f ( x ) dx = 1 −∞ b 3. P ( a ≤ X = ≤ b) ( x ) dx ∫ f= area under f ( x ) from a to b for any a and b a Note: For a continuous random variable X , the following are true: ) 0 1. P ( X= x= 2. P ( x1 ≤ X ≤ x2 )= P ( x1 < X ≤ x2 )= P ( x1 ≤ X < x2 )= P ( x1 < X < x2 ) (4.2) Cumulative Distribution Functions Definition: The cumulative distribution function (CDF) of a continuous random variable X is x F ( x= ) P ( X ≤ x =) ∫ f ( u ) du −∞ PDF from the CDF: If the derivative of F ( x ) exists, d f ( x ) F= = '( x)  F ( x ) . dx  (4.3) Mean and Variance of a Continuous Random Variable -Mean and Variance- Suppose that X is a continuous random variable with probability density function f ( x ). The mean or expected value of X , denoted as µ or E ( X ) , is ∞ µ E ( X= = ) ∫ x ⋅ f ( x ) dx −∞ The variance of X , denoted as V ( X ) or σ 2 , is ∞ ∞ σ2 = V (X ) = ∫ (x − µ) ⋅ f ( x ) dx = ⋅ f ( x ) dx − µ 2 2 ∫x 2 −∞ −∞ The standard deviation of X is σ = σ 2 -Expected Value- Expected Value of a Function of a Continuous Random Variable: If X is a continuous random variable with probability density function f ( x ) , ∞ ( X ) E  h= ∫ h ( x ) ⋅ f ( x ) dx −∞  Problems that f ( x ) e ( ) for 4 < x. Determine the following: − x−4 1) Suppose= a) P (1 < x ) b) P ( 5 < x ) c) P ( 4 < x ≤ 8 ) d) x such that P ( X < x ) = 0.90 2) Suppose that the cumulative distribution function of the random variable X is 0 x < −2  ( x ) 0.25 x + 0.5 −2 ≤ x < 2 F= 1 x≥2  Determine the following: a) f ( x ) b) P ( X < 1.8 ) c) P ( X > −1.5 ) d ) P ( X < −2 ) e) P ( −1 < X < 1)  = 3) Suppose that f ( x ) 0.125 x for 0 < x < 4. a) Determine the mean and variance of X. Y 3 X + 1 , find E (Y ) and V (Y ). b) If = 70 = 4) Suppose that f ( x) for 1 < x < 70. Determine the following: 69 x 2 a) P ( X > 50 ) b) The mean and variance of X. c) If Y = 2.50 X , find E (Y ) and V (Y ).