STAT/MATH 395 PROBABILITY II

Transcription

1 STAT/MATH 395 PROBABILITY II Continuous Random Variables Néhémy Lim University of Washington Winter 2017

2 Outline Probability Density Function and Cumulative Distribution Function Expectation and Variance Common Continuous Distributions Uniform Distribution Gaussian Distribution Quantiles Exponential Distribution

3 Outline Probability Density Function and Cumulative Distribution Function Expectation and Variance Common Continuous Distributions Uniform Distribution Gaussian Distribution Quantiles Exponential Distribution

4 Example Let X be the distance between Federer s position when he serves and the location where the ball lands. What is the probability that X is exactly 15 metres? Hint: See Hawk eye system

5 Definition (Probability density function) A rrv is said to be (absolutely) continuous if there exists a real-valued function f X such that, for any subset B R: P(X B) = f X (x) dx (1) Then f X is called the probability density function (pdf) of the random variable X. Property If X is a continuous rrv, then For any real numbers a and b, with a < b B P(a X b) = P(a X < b) = P(a < X b) = P(a < X < b) (2) P(X = a) = 0, for all a R

6 Theorem A probability density function completely determines the distribution of a continuous real-valued random variable.

7 Definition (Cumulative distribution function) Let (Ω, A, P) be a probability space. The (cumulative) distribution function (cdf) of a real-valued random variable X is the function F X given by F X (x) = P(X x), for all x R (3) Property Let F X be the cumulative distribution function of a random variable X. Following are some properties of F X : F X is increasing : x y F X (x) F X (y) lim x F X (x) = 1 and lim x F X (x) = 0 F X is càdlàg : FX is right continuous : lim x x0 F X (x) = F X (x 0 ), for x 0 R FX has left limits : lim x x0 F X (x) exists, for x 0 R

8 Cumulative Distribution Function of a continuous rrv Property Let X be a continuous rrv with pdf f X. Then the cumulative distribution function F X of X is given by : F X (x) = x f X (t) dt (4) Corollary Let X be a continuous rrv with pdf f X and cdf F X. Then F X (x) = f X(x), if f X is continuous at x; P(a X b) = F X (b) F X (a), for any a, b R with a < b; A cdf completely determines the distribution of a continuous real-valued random variable.

9 Proposition Let X be a continuous rrv on probability space (Ω, A, P) with pdf f X. Then, we have : f X is nonnegative on R : f X is integrable on R and f X (x) 0, for all x R; (5) f X (x) dx = 1 (6) Definition (Valid probability density function) A real-valued function f is said to be a valid pdf if it satisfies (5) and (6).

10 Example A continuous rrv X is said to follow a uniform distribution on [0, 1/2] if its pdf is : f X (x) = { c if 0 x 1/2 0 otherwise = c1 [0,1/2] (x) Questions. (1) Graph f X (2) Determine c such that f X satisfies the properties of a pdf. (3) Give the cdf of X and graph it.

11 Example Let X be the duration of a telephone call in minutes and suppose X has pdf : f X (x) = c e x/10 1 [0, ) (x) Questions. (1) Which value(s) of c make(s) f X a valid pdf? (2) Find the probability that the call lasts less than 5 minutes.

12 Outline Probability Density Function and Cumulative Distribution Function Expectation and Variance Common Continuous Distributions Uniform Distribution Gaussian Distribution Quantiles Exponential Distribution

13 Definition (Expected Value) Let X be a continuous rrv with pdf f X. If x f X(x) dx <, then the mathematical expectation (or expected value or mean) of X exists and is defined as follows : E[X] = xf X (x) dx (7) Let g : R R be a piecewise continuous function. If random variable g(x) is integrable. Then, the mathematical expectation of g(x) exists and is defined as follows : E[g(X)] = g(x)f X (x) dx (8) Example. Expectation of a uniform distribution on [0, 1/2]

14 Property Let X be a continuous rrv with pdf f X. for all c R, E[c] = c If g : R R is a nonnegative piecewise continuous function and g(x) is integrable. Then, we have : E[g(X)] 0 (9) If g 1 : R R and g 2 : R R are piecewise continuous functions and g 1 (X) and g 2 (X) are integrable such that g 1 g 2. Then, we have : E[g 1 (X)] E[g 2 (X)] (10)

15 Linearity of Expectation Property Let X be a continuous rrv with pdf f X. If c 1, c 2 R and g 1 : R R and g 2 : R R are piecewise continuous functions and g 1 (X) and g 2 (X) are integrable. Then, we have : E[c 1 g 1 (X) + c 2 g 2 (X)] = c 1 E[g 1 (X)] + c 2 E[g 2 (X)] (11)

16 Variance Standard Deviation Definition Let X be a rrv on probability space (Ω, A, P). If E[X 2 ] exists, the variance of X is defined as follows : Var(X) = E[(X E[X]) 2 ] (12) The standard deviation of X is: σ X = Var(X) (13)

17 Property Let X be a real-valued random variable. Var(X) 0 If a, b R are two constants, then Var(aX + b) = a 2 Var(X) (König-Huygens formula) When E[X 2 ] exists : Var(X) = E[X 2 ] (E[X]) 2 (14) If X is a continuous rrv with pdf f X, Equation (14) becomes : Var(X) = ( ) 2 x 2 f X (x) dx xf X (x) dx Example. Variance of a uniform distribution on [0, 1/2]

18 Outline Probability Density Function and Cumulative Distribution Function Expectation and Variance Common Continuous Distributions Uniform Distribution Gaussian Distribution Quantiles Exponential Distribution

19 Uniform Distribution A continuous rrv is said to follow a uniform distribution U(a, b) on a segment [a, b], with a < b, if its pdf is f X (x) = 1 b a 1 [a,b](x) (15) If X follows a uniform distribution U(a, b), then E[X] = a + b 2 (16) Var(X) = (b a)2 12 (17)

20 Motivation Generating random numbers based on a probability density function

21 Example A restaurant wants to advertise a new burger they call The Quarter-kilogram.

22 Normal distribution A continuous random variable is said to follow a normal (or Gaussian) distribution N (µ, σ 2 ) with parameters, mean µ and variance σ 2 if its pdf f X is given by: f X (x) = 1 { σ 2π exp 1 ( ) } x µ 2, for x R (18) 2 σ If X follows a normal distribution N (µ, σ 2 ), then E[X] = µ (19) Var(X) = σ 2 (20)

23 Property If X follows a normal distribution N (µ, σ 2 ). Then its pdf f X has the following properties: (1) f X is symmetric about the mean µ : (2) f X is maximized at x = µ. f X (µ x) = f X (µ + x), for x R (21) (3) The limit of f X (x), as x approches or, is 0: lim x f X(x) = lim x f X(x) = 0 (22)

24 Standard Normal Distribution Definition We say that a continuous rrv X follows a standard normal distribution if X follows a normal distribution N (0, 1) with mean 0 and variance 1. The cdf of a standard normal random variable is denoted Φ, that is : Φ(x) = x 1 2π e t2 2 dt, for x R (23) Property The cdf of a standard normal random variable satisfies the following property: Φ( z) = 1 Φ(z), for z R (24)

25 Property Let a R, a 0 and b R. If X follows a normal distribution N (µ, σ 2 ), then random variable ax + b follows a normal distribution N (aµ + b, a 2 σ 2 ). Corollary If X follows a normal distribution N (µ, σ 2 ), then random variable Z defined by: Z = X µ σ is a standard normal random variable.

26 Finding Normal probabilities. If X N (µ, σ 2 ). In order to find probabilities P(a X b) : (1) Transform X, a, and b, by: Z = X µ σ (2) Use the standard normal N (0, 1) Table to find the desired probability. Example. Let X be the weight of a burger. Assume X follows a normal distribution with mean 250 grams and standard deviation 15 grams. (a) What is the probability that a randomly selected burger has a weight below 240 grams? (b) What is the probability that a randomly selected burger has a weight above 270 grams? (c) What is the probability that a randomly selected burger has a weight between 230 and 265 grams?

27 Example Suppose X, the grade on a midterm exam, is normally distributed with mean 70 and standard deviation 10.The instructor wants to give 15% of the class an A. What cutoff should the instructor use to determine who gets an A?

28 Definition (Quantile) Let X be a rrv with cumulative distribution function F X and α [0, 1]. A quantile of order α for the distribution of X, denoted q α, is defined as follows : Remarks : q α = inf {x R F X (x) α } (25) The quantile of order 1/2 is called the median of the distribution If F X is a bijective function, then q α = F 1 X (α) Property For α (0, 1), the quantile function of a standard normal random variable is given by : Φ 1 (α) = Φ 1 (1 α) (26)

29 Finding the Quantiles of a Normal Distribution. In order to find the value of a normal random variable X N (µ, σ 2 ): (1) Find in the Table the z = (x µ)/σ value associated with the desired probability. (2) Use the transformation x = µ + zσ. Example. Suppose X, the grade on a midterm exam, is normally distributed with mean 70 and standard deviation 10. (a) The instructor wants to give 15% of the class an A. What cutoff should the instructor use to determine who gets an A? (b) The instructor now wants to give the next 10% of the class an A-. For which range of grades should the instructor assign an A-?

30 Motivating example Let Y be the discrete rrv equal to the number of people joining the line to visit the Eiffel Tower in an interval of one hour. Assume that the mean number of people arriving in an interval of one hour is 800. Let X be the waiting time until the first visitor arrives.

31 Exponential distribution A continuous random variable is said to follow an exponential distribution E(λ) with λ > 0 if its pdf f X is given by: f X (x) = λe λx 1 [0, ) (x), for x R (27) If X follows an exponential distribution E(λ), then E[X] = 1 λ (28) Var(X) = 1 λ 2 (29) Example. Suppose that people join the line to visit the Eiffel Tower at a mean rate of 800 visitors per hour. What is the probability that nobody joins the line in the next 30 seconds?