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Random Variables

Random Variables. Suppose we toss a coin 3 times. Then our sample space is :. S = {. HHH,. HHT, HTH, THH,. HTT, THT, TTH,. TTT }. What if we are interested in the number of heads that appear in our experiment ?. We will denote this variable by X. ( So, X represents the number of heads ).

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Random Variables

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  1. Random Variables Suppose we toss a coin 3 times. Then our sample space is : S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } What if we are interested in the number of heads that appear in our experiment ? We will denote this variable by X ( So, X represents the number of heads ) In this example, X could be : 0, 1, 2, or 3 If the outcome of the experiment is HTH, then X = 2 X is called a random variable

  2. Random Variables • A random variable is a variable whose value is a numerical • outcome of a random phenomenon. • These are usually denoted by a capital letter • We are interested in random variables such as • the mean of a random sample. There are two types of we will study intently : • Discrete random variables • Continuous random variables

  3. Discrete Random Variables Value of X Probability A discrete random variable X has a finite number of possible values. The probability distribution of X lists the values and their probabilities. . . . x1 x2 x3 xk . . . p1 p2 p3 pk The probabilities pi must satisfy two requirements : 1) Every pi is between 0 and 1 2) p1 + p2 + p3 + … + pk = 1 Find the probability of an event by adding the pi’s of the particular xi’s that make up the event.

  4. Discrete Random Variables Player Probability J G E K 2 1 0 3 Example : Jerry, Elaine, George, and Kramer are going to have a contest. They are going to see who can be the master of their domain for the longest amount of time. E G J K 0.42 0.31 0.24 0.03 We can assign each Player a value.

  5. Discrete Random Variables Player Probability Example : Jerry, Elaine, George, and Kramer are going to have a contest. They are going to see who can be the master of their domain for the longest amount of time. 0 1 2 3 0.42 0.31 0.24 0.03 Q: What is the probability that George or Kramer will win? A: P(Winner is 1 or 3) = P(X = 1) + P(X = 3) = 0.31 + 0.03 = 0.34

  6. Probability Histogram 0.5 0.4 Probability 0.3 0.2 0.1 0 1 2 3 Outcome

  7. Example : Go back to the experiment of tossing a coin 3 times. S = { HHH, HHT, HTH, THH, TTH, THT, HTT, TTT } Q: What is P(TTH) ? 1/2 * 1/2 * 1/2 = 1/8 Note: This is an equally likely experiment. Let X be the amount of heads that appears in our experiment. Q: What is the probability of getting no heads ? A: P(No Heads) = P(X = 0) = 1/8

  8. Example : Go back to the experiment of tossing a coin 3 times. S = { HHH, HHT, HTH, THH, TTH, THT, HTT, TTT } Q: What is P(TTH) ? 1/2 * 1/2 * 1/2 = 1/8 Note: This is an equally likely experiment. Let X be the amount of heads that appears in our experiment. Q: What is the probability of getting two heads ? A: P(2 Heads) = P(X = 2) = P(HHT) + P(HTH) + P(THH) = 1/8 + 1/8 + 1/8 = 3/8

  9. Continuous Random Variable Newman comes up to you and asks you to play a game. He has picked a number between 0 and 1, and he wants you to try and guess the number he has picked. If we let Newman’s number be represented by X, what is the probability you will guess his number ? In other words, what is P(guess = X) ? 1 /  = 0 To be fair, we should assign you a range of numbers, say 0.3 through 0.7 This gives us : P( 0.3 < X < 0.7 ) Q: How do we find this probability ?

  10. Continuous Random Variable Definition : A continuous random variable X takes all values in an interval of numbers. • The probability assigned to an event can be found by • assigning an area under a density curve. • The probability distribution of X is described by a density • curve. The probability of an event is the area under the • curve and above the values of X that make up the event. Definition : A uniform distribution is a distribution of constant height.

  11. Uniform Distributions Example: Draw a uniform distribution over the interval from 0 to 4 : 0.25 0 1 2 3 4

  12. Uniform Distributions 0 1 Go back to Newman’s example: • This is a uniform distribution over the interval • from 0 to 1

  13. 0.3 0.7 0 1 We assigned ourselves an interval to guess Newman’s number We used the interval from 0.3 to 0.7 What is the probability we are correct ? P( 0.3 < X < 0.7 ) = 0.4 the shaded area =

  14. 0.2 0.4 0.8 0 1 What if we were given the ranges from 0 to 0.2 and from 0.4 to 0.8 ? What is the probability that we have covered Newman’s pick? P( 0 < X < 0.2 or 0.4 < X < 0.8) = 0.2 + 0.4 = 0.6

  15. Non-Uniform Distributions So, what if we don’t have a uniform distribution ? We can solve these if we have a normal distribution.

  16. Normal Distributions as Probability Distributions X -  Z =  Recall that N( , ) represents a normal distribution with mean  and standard deviation . To get a standard score ( z-score) : By standardizing our scores, we go from N( , ) to N(0, 1)

  17. 4.50 Example : The Soup Nazi charges, on the average, $4.50 for a cup of soup, and if you’re lucky, some bread, with a standard deviation of $0.45. What is the probability that our check will be more than $5.00 ?

  18. 0.1335 0.8665 4.50 5.00 - 4.50 Z = 0.45 5.00 What is the probability that our check will be more than $5.00 ? P (X > 5 ) = 0.1335 = 13.35 % = 1.11

  19. Homework 41, 42, 44, 46, 47, 49, 50, 53, 54, 55

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