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Chapter 3, Random Variables and Probability Distributions 3.1 Random variables (1) An example: - How to calculate the average grade of a student? - Use random variable, the example: what is the grades of a student marks comments r. v. (X) F failed 0 D, C passed 1 B good 2 A excellent 3 (2) Definition: - definition of random variable (r. v.): for an experiment with sample space S, a function X assigning to every element s S, which is a real number, is called a random variable. - illustration X(x) x r. v. space sample space S - note that the assignment is not unique. Many assignments can work equally well. - discrete r.v. and continuous r. v. discrete: assigning to countable outcomes continuous: assignment to continuous outcomes 3.2 Discrete probability distributions (1) definition: the set of ordered pairs of (x, f(x)) is a probability function of discrete r. v. if for each possible outcome x: (a) f(x) 0 f (x) 1 (b) all x (c) P(X = x) = f(x) (2) the cumulative distribution F(x) P(X x) f (t), for x t x (3) An example: in the above example, assume: marks r. v. (X) probability F 0 0.05 3-1 D, C B A 1 2 3 0.25 0.5 0.2 then: f(3) = P(X = 3) = 0.2 F(X) = P(X 1) = 0.25 + 0.5 + 0.2 = 0.95 another way to calculate: F(X 1) = 1 - P(X = 0) = 1 - 0.05 = 0.95 3.3 Continuous probability distribution (1) When X is a continuous r. v., f(x) is a probability density function, which satisfy: (a) f(x) 0, for all x (b) f (x)dx 1 b (c) P(a X b) f (x)dx a (2) The cumulative distribution F(x) P(X x) x f (t)dt , for - < x < - note: P(a < X < b) = F(b) - F(a) dF(x) f (x) dx F(x) = 0 3.4 Empirical distributions (1) Empirical distribution is a probability model that often used in data analysis (2) How to obtain empirical distribution: the example the data (number of hours of exercise per week of the 12 students) the stem-leaf plot stems 00 02 04 06 08 10 leaves 0,0 4, 4, 4, 5, 5 6, 7 8 0, 0 frequency 2 0 5 2 1 2 the histogram estimating the probability distribution 3-2 relative frequency 1/6 0 5/12 1/6 1/12 1/6 3.5 Joint probability distributions (1) The example: what is the probability of picking a student who is a female and exercises 5 hours per week? (2) The definition of joint probability (for discrete r. v.): the function f(x, y) is a joint probability distribution of the discrete r. v. X and Y if (a) f(x, y) 0, for all (x, y) (b) f ( x, y ) 1 x y (c) P(X = x, Y = y) = f(x, y) - For any region A in the xy plane, P[(X,Y) A] f (x,y) A (3) The example: - the definition of the r. v. X2: 0 - Male 1 - Female X4: 0 - 0 hours 1 - 4 hours 2 - 5 hours 3 - 6 hours 4 - 7 hours 5 - 8 hours 6 - 10 hours - the joint probability distribution X4 \ X2 0 (M) 0 (0) 1/12 1 (4) 3/12 2 (5) 0 3 (6) 1/12 4 (7) 0 5 (8) 1/12 6 (10) 2/12 sum 8/12 - 1(F) 1/12 0 2/12 0 1/12 0 0 4/12 sum 2/12 3/12 2/12 1/12 1/12 1/12 2/12 1 therefore: P(X2 = 1, X4 = 2) = 2/12 = 1/6 (4) Another example: the probability of male students who exercise more than 5 hours (X4 > 2) is: P(X1 = 0, X4 > 2) = (1 + 0 + 1 + 2)/12 = 4/12 = 1/3 (5) Marginal distribution: - Question: given a female student is picked, what is the probability that she exercise 5 hours per week? 3-3 - Solution: marginal distribution Definition: the marginal distributions of X alone (or Y alone) is: g(x) = y f(x, y) (or h(y) = x f(x, y)) - In the above example, the marginal probability distribution of X2 is (the column sum): g(X2) = {8/12, 4/12} = {2/3, 1/3} likewise, the marginal probability of X4 is the row sum. (6) Conditional distribution - The conditional distribution: given that X = x, the conditional distribution of Y is: f(y / x) = f(x, y) / g(x), g(x) > 0 - - In the above example, given X2 = 1, the condition probability distribution of X4 is: f(X4 / X2 = 1) = f(0, 1)/g(1) (1/12)/(4/12) = 1/4 f(1, 1)/g(1) (0)/(4/12) = 0 f(2, 1)/g(1) (2/12)/(4/12) = 1/2 f(3, 1)/g(1) (0)(4/12) = 0 f(4, 1)/g(1) (1/12)(4/12) = 1/4 f(5, 1)/g(1) (0)(4/12) = 0 f(6, 1)/g(1) (0)(4/12) = 0 Therefore, P(X4 = 2 / X2 = 1) = f(2, 1)/g(1) = 1/2. Note that both the marginal distribution and the conditional probability distribution are distributions. Based on these distributions we can further calculate the probability. The continuous probability distribution: the function f(x, y) is a join density function of the continuous random variables X and Y if (a) f(x, y) 0, for all (x, y) f (x, y)dxdy 1 (c) P[(X,Y) A] f (x,y)dxdy (b) A - note that the marginal distribution and the conditional distribution are defined in a similar way as in the discrete case. In addition, all the calculations are similar. (7) Multiple variable case: similar to the two variable case. 3-4