Probability Theory: Continuous Random Variables and Distributions, Study notes of Probability and Statistics

Download Probability Theory: Continuous Random Variables and Distributions and more Study notes Probability and Statistics in PDF only on Docsity! MATH 243, LECTURE 12 1. Probability distributions To be systematic about probabilities in the continuous setting, we have to formalize the notion of distribution. Definition 1. (1) A distribution for a continuous random variable can be any non-negative function D where the total area under the function is 1. (2) If X is distributed according to a distribution D then P (a ≤ X ≤ b) is the area between the line x = a, the line x = b, the graph of D, and the x-axis. (3) The sample space for such a variable is all x where D is non-zero. (4) If D is equal to either 0 or one other fixed number, we say D is a uniform distribution. Example 2. Revisit the example about numbers between 0 and 3. • Our sample space is the closed interval [0, 3]. • Our probability distribution is the function p(t) = { 1/3 0 ≤ t ≤ 3 0 else Again, what is the probability that some randomly chosen number is between 1.5 and 2.4? Etc. Example 3. Which of the following are distributions? • f(x) = 1 for x between −2 and −1, and zero otherwise. • f(x) = 1 over the whole real number line. • f(x) = x2 for x between 0 and 2 and zero otherwise. • f(x) = x2 for x between −1 and 1 and zero otherwise. Example 4. Suppose that X is distributed according to D =  1 12 1 ≤ x ≤ 3 1 2 3 ≤ x ≤ 4 1 9 4 ≤ x ≤ 7 0 otherwise • What is the state space? • What is P (2 ≤ X ≤ 3)? • Which is more likely, that X is between 3 and 4 or that X is between 5 and 7? Compare the answer to the last question to the answer you would get if X were uniformly distributed. One way to get a distribution is to take the histogram of a data set and divide by the total number of individuals so that the area is one. The probabilities computed correspond to percentages of the data. Example 5. Translate between probability and percentile questions about scores of students taking an exam. 1 2 MATH 243, LECTURE 12 2. Normal distributions Example 6. [Main Example] Z is a number chosen with normal distribution N(µ, σ). We’ll use N(0, 1) as our first example. • Our random event is the choice of a number. • Our random variable Z is the value of the number. • Our sample space is all possible real numbers. • Our probability distribution is the function e−t 2/2/ √ 2π. • What is P (Z > 3)? (Hint: use 68-95-99.7 rule.) • What is P (0 ≤ Z ≤ 2)? • P (Z > 100)? Doing calculations with normal distributions works much as before. Instead of calculating percentiles, we are calculating probabilities. Example 7. Let our random event be the random choice of a U.S. adult male, assuming a probability distribution which is approximately normal N(70, 4).. (1) What is our random variable? Our state space? (theoretical vs. actual) (2) What is the probability that our randomly chosen man is between 70 inches and 73 inches tall? 3. Sampling distributions A basic practice in statistics is to take a distribution and sample from it. Definition 8. Given a population, the sampling distribution for samples of size n is the probability distri- bution of some parameter as we take values n times. We go thoroughly through an example looking at sampling distributions and comment thoroughly as we go along. Consider the collection of numbers S = {2, 2, 5, 6, 7}. What is the mean? Ans: 4.4. We can consider all samples from our collection of size 1. There are 5 of them and again, the mean of the samples is again 4.4. The standard deviation is 2.302. We can now consider all samples from S of size 2, and take the mean of each sample. Our samples are {{2, 2}, {2, 5}, {2, 6}, {2, 7}, {2, 5}, {2, 6}, {2, 7}, {5, 6}, {5, 7}, {6, 7}}. Our means are {2, 3.5, 4, 4.5, 3.5, 4, 4.5, 5.5, 6, 6.5} Notice that the spread is smaller, even though there are more numbers. The mean is still 4.4, and the standard deviation is now 1.329. The probability distribution of these numbers is an example of sampling distribution. In this case it is for the mean of a sample of 2 from the set S. There are 7 possible outcomes which don’t have equal probabilities. X 2 3.5 4 4.5 5.5 6 6.5 Probability .1 .2 .2 .2 .1 .1 .1