Continuous Probability Distributions Lec T7 Sum 19 3339C 3

Determine the cdf of a Uniform Distribution Let X ∼ U(0, 5) such that the pdf of X is: ( 1 0≤x ≤5 f (x) = 5 0 otherwise Find the cdf F (x)for X . 24 / 53 0.6 0.4 0.2 0.0 Cumulative Probability 0.8 1.0 Cumulative Density Function 0 1 2 3 4 5 X 25 / 53 Using the cdf F (X ) to Compute Probabilities Let X be a continuous random variable with pdf f (x) and cdf F (x). Then for any number a, P(X > a) = 1 − F (a) and for any two numbers a and b with a < b, P(a ≤ X ≤ b) = F (b) − F (a) 26 / 53 Example Using CDF Suppose we have a cdf; F (x) =   0, x 3 +1 ,  9  1, x ≤ −1 −1 ≤ x < 2 x ≥ 2. 1. Determine P(X ≤ 0) 2. Determine P(0 < X ≤ 1) 3. Determine P(X ≥ 0.5) 4. Given this CDF determine the pdf f (x). 27 / 53 Example Suppose we have a pdf of ( f (x) = 3 2 8x 0 0≤X ≤k otherwise a) Determine k . b) Give the cdf of this distribution. c) Determine x0 such that P(X ≤ x0 ) = 0.125 28 / 53 Quantiles Let F be a given cumulative distribution and let p be any real number between 0 and 1. The (100p)th percentile of the distribution of a continuous random variable X is defined as F −1 (p) = min{x|F (x) ≥ p}. For continuous distributions, F −1 (p) is the smallest number x such that F (x) = p. 29 / 53 Determine the Percentiles Given a cdf, F (x) =   0 1 3 x 8  1 X <0 0≤X ≤2 X >2 1. Determine the 90th percentile. 2. Determine the 50th percentile. 3. Find the value of c such that P(X ≤ c) = 0.75. 30 / 53 Expected Values for Continuous Random Variables The expected or mean value of a continuous random variable X with pdf f (x) is Z ∞ E(X ) = xf (x)dx. −∞ More generally, if h is a function defined on the range of X , Z ∞ E(h(X )) = h(x)f (x)dx. −∞ 31 / 53 Example The following is a pdf of X , ( f (x) = 3 2 (1 0 − x 2) 0 ≤ X ≤ 1 otherwise 1. Determine E(X ). 2. Determine E(X 2 ) 32 / 53