Lecture 22 - Delay Time of a Flight Example

University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

60

Pages:

4
Author:

Niklas

LECTURE 22 Delay Time of a Flight About the delay time of a flight, we might observe the following data and plot them using a histogram: Delay Frequency (0, 5] 0.633 (5, 10] 0.231 (10, 15] 0.086 (15, 20] 0.043 (20, 25] 0.000 (25, ∞] 0.000 Assuming that we have infinite observations, we can increase the number of classes and approximate the histogram with a curve: probability density curve. Probability Density Function The probability density function is a real-valued function assuming non-negative values: f(x) = (1/5) * e^(-x/5) The interpretation of the density function is analogous to the histogram: the areas represent probabilities rather than frequencies. x ∈ [0, ∞] The probability of observing a value between 5 and 10: P(5 x = 10) = ∫(5 to 10) f(t) dt = ∫(5 to 10) (1/5) * e^(-t/5) dt Calculations The calculations for the integral are shown step-by-step. A Relevant Quantity A relevant quantity is: P(x <= 10) = ∫(0 to 10) f(x) dx = ∫(0 to 10) (1/5) * e^(-t/5) dt The calculations for this integral are also shown step-by-step. Conclusion The final result is: P(x = 10) = 1 - (1/e^2) ≈ 0.86 Probability Density Function The probability density function of a continuous random variable X is a non-negative function f(x) whose area under the curve in a range of values is the probability of X in that range: ∫(a to b) f(t) dt = P(a = X = b) The probability density function satisfies the following properties: f(x) = 0 ∫(-∞ to ∞) f(t) dt = 1 Cumulative Distribution Function The cumulative distribution function of a variable X is the function: F(x) = P(X = x) = ∫(-∞ to x) f(t) dt and satisfies the following properties: F(x) = 0 F(x) is not decreasing lim(x - -∞) F(x) = 0 lim(x - +∞) F(x) = 1 Note that: P(a = X = b) = F(b) - F(a) Example There are several density functions. Any non-negative function that integrates to 1 is a density function. Uniform between 0 and 3 f(x) = { 1/3 for 0 = x = 3 { 0 otherwise Cumulative Distribution Function F(x) = ∫(0 to x) λe^(-λt) dt = -e^(-λt) |_(0 to x) = 1 - e^(-λx)