Random Variables Random Variables A random variable usually

Random Variables

Random Variables A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon example Flip a coin three times; X = the total number of heads. The values o X are. X= 0, 1, 2, 3 Throw two dice; X = the sum of the numbers facing up. The values of X are. X= 2, 3, 4, …, 12 Throw one die over and over until you get a six; X = the number of X=1, 2, 3, 4, . . .

Types of Random Variables Discrete random variables are ones that have a finite or countable number of possible outcomes (like number of heads when flipping several coins). Continuous random variables are ones that have an infinite or uncountable number of possible outcomes (like your exact speed on the highway, or how far someone jumps)

Probability with discrete variables Throw a pair of fair dice, and take X to be the sum of the numbers facing up. X= 2, 3, 4, … , 12 ◦ The event that X = 2 is {(1, 1)} The event that you throw a 2 ◦ The event that X = 3 is {(2, 1), (1, 2)} The event that you throw a 3 ◦ The event that X = 4 is {(3, 1), (2, 2), (1, 3)} The event that you throw a 4 ◦ P(X = 4) = 1/12 ◦ The probability that X = 4 is 1/12

practice Throw a pair of fair dice, and take X to be the sum of the numbers facing up. X= 2, 3, 4, … , 12 P(X=1) = 0 P(X=2) = 1/36 P(X=3) = 2/36 P(X=4) = 3/36 Problem 2: Tossing a coin 4 times. Find the discrete probability of the following: let X= number of heads P(X=0) = 1/16 P(X=1) = 4/16 P(X=2) = 6/16 P(X=3) = 4/16

Problem 2: Tossing a coin 4 times. Find the probability of the following: Where: X= number of heads Number of heads Probability 0 1 2 3 4 1/16 4/16 6/16 4/16 1/16 What is: P(X ≤ 2) = P(x=0) + P(x=1) + P(x=2) = 1/16 + 4/16 + 6/16 = 11/16 Or 0. 6875

Education Level: A study of education followed a large group of fifthgrade children to see how many years of school they eventually completed. Let X be the highest year of school that a randomly chosen 5 th graders completes Years probability 4 5 6 7 8 9 10 11 12 0. 010 0. 007 0. 013 0. 032 0. 068 0. 070 0. 041 0. 752 A. Is this a “legit” continuous probability distribution? Yes, because the probabilities add up to 1 B. What is the percent of 5 th graders eventually finished 12 th grade? There are 75. 2% 5 th graders eventually finished 12 th grade. C. Find P(X≥ 6) C. Find P(X>6) 0. 983 0. 931