The Median of a Continuous Distribution To calculate

The Median of a Continuous Distribution To calculate the median of a continuous distribution, we must use the cumulative distribution function F(x). The probability that X has a lower value than the median is 0. 5. P(X ≤ x 0. 5) = F(x 0. 5) = 0. 5 F(x) 1 0. 5 0 a x 0. 5 b x

The continuous random variable X is distributed with cumulative distribution function F where F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4 Find the median F(x 0. 5) = 0. 5

It is also possible to use the cumulative distribution function F(x) in order to calculate the inter-quartile range and the inter-percentile range of the distribution. The inter-quartile range The probability that X has a lower value than the lower quartile is 0. 25. P(X ≤ x 0. 25) = F(x 0. 25) = 0. 25 F(x) 1 The probability that X has a lower value than the upper quartile is 0. 75. P(X ≤ x 0. 75) = F(x 0. 75) = 0. 75 F(x) 1 0. 75 0. 25 0 a x 0. 25 b x 0 a x 0. 75 b x

The continuous random variable X is distributed with cumulative distribution function F where F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4 Calculate the inter-quartile range. F(x 0. 75) - F(x 0. 25) F(x 0. 75) = Inter-quartile range = F(x 0. 25) = 3. 63 – 2. 52 = 1. 11

The inter-percentile range The continuous random variable X is distributed with cumulative distribution function F where F(x) = 0 for x < 0 F(x) = for 0 ≤ x ≤ 4 F(x) = 1 for x > 4 Calculate the inter-percentile range 10 -90 F(x 0. 9) - F(x 0. 1) F(x 0. 9) = F(x 0. 1) = The inter-percentile range 10 -90 = 3. 86 – 1. 86 = 2

Percentile Exercise Homework 13