Random Variables Learn how to characterize the pattern of the distribution of values that a random variable may have, and how to use the pattern to find probabilities Budhi Setiawan Teknik Sipil - UNSRI

2 What is a Random Variable? Random Variable: an outcome or event may be identified through the value(s) of a function, which usually denoted with a capital letter Two different broad classes of random variables: 1.A continuous random variable can take any value in an interval or collection of intervals. 2.A discrete random variable can take one of a countable list of distinct values. If the value of X represent flood above mean level, then X > 7 meter stand for the occurrence of floods above 7 meter

3 Random factors that will determine how enjoyable the event is: Temperature: continuous random variable (any value, integer or decimal) Number of airplanes that fly overhead: discrete random variable (integer only) Example: Random Variables at an Outdoor Graduation or Wedding

4 What is the probability that three tosses of a fair coin will result in three heads? Assuming boys and girls are equally likely, what is the probability that 3 births will result in 3 girls? Assuming probability is 1/2 that a randomly selected individual will be taller than median height of a population, what is the probability that 3 randomly selected individuals will all be taller than the median? Example: Random Variables:Probability an Event Occurs 3 Times in 3 Tries Answer to all three questions = 1/8. Discrete Random Variable X = number of times the “outcome of interest” occurs in three independent tries.

5 Discrete Random Variables Discrete random variable: can only result in a countable set of possibilities – often a finite number of outcomes, but can be infinite. X the random variable. k = a number the discrete r.v. could assume. P(X = k) is the probability that X equals k. Example: It’s Possible to Toss Forever Repeatedly tossing a fair coin, and define: X = number of tosses until the first head occurs Any number of flips is a possible outcome. P(X = k) = (1/2) k

6 Probability Distribution of a Discrete R.V. Using the sample space to find probabilities: Step 1: List all simple events in sample space. Step 2: Find probability for each simple event. Step 3: List possible values for random variable X and identify the value for each simple event. Step 4: Find all simple events for which X = k, for each possible value k. Step 5: P(X = k) is the sum of the probabilities for all simple events for which X = k. Probability distribution function (pdf) X is a table or rule that assigns probabilities to possible values of X.

7 Example:How Many Girls are Likely? Family has 3 children. Probability of a girl is ? What are the probabilities of having 0, 1, 2, or 3 girls? Sample Space: For each birth, write either B or G. There are eight possible arrangements of B and G for three births. These are the simple events. Sample Space and Probabilities: The eight simple events are equally likely. Random Variable X: number of girls in three births. For each simple event, the value of X is the number of G’s listed.

8 Example: How Many Girls? (cont) Probability distribution function for Number of Girls X: Value of X for each simple event: Graph of the pdf of X:

9 Conditions for Probabilities for Discrete Random Variables Condition 1 The sum of the probabilities over all possible values of a discrete random variable must equal 1. Condition 2 The probability of any specific outcome for a discrete random variable must be between 0 and 1.

10 Cumulative Distribution Function of a Discrete Random Variable Cumulative distribution function (cdf) for a random variable X is a rule or table that provides the probabilities P(X ≤ k) for any real number k. Cumulative probability = probability that X is less than or equal to a particular value. Example: Cumulative Distribution Function for the Number of Girls (cont)

11 Finding Probabilities for Complex Events Example: A Mixture of Children pdf for Number of Girls X: What is the probability that a family with 3 children will have at least one child of each sex? If X = Number of Girls then either family has one girl and two boys (X = 1) or two girls and one boy (X = 2). P(X = 1 or X = 2) = P(X = 1) + P(X = 2) = 3/8 + 3/8 = 6/8 = 3/4

12 Expectations for Random Variables The expected value of a random variable is the mean value of the variable X in the sample space, or population, of possible outcomes. If X is a random variable with possible values x 1, x 2, x 3,..., occurring with probabilities p 1, p 2, p 3,..., then the expected value of X is calculated as

13 Standard Deviation for a Discrete Random Variable The standard deviation of a random variable is essentially the average distance the random variable falls from its mean over the long run. If X is a random variable with possible values x 1, x 2, x 3,..., occurring with probabilities p 1, p 2, p 3,..., and expected value E(X) = , then

14 Binomial Random Variables 1. There are n “trials” where n is determined in advance and is not a random value. 2. Two possible outcomes on each trial, called “success” and “failure” and denoted S and F. 3. Outcomes are independent from one trial to the next. 4. Probability of a “success”, denoted by p, remains same from one trial to the next. Probability of “failure” is 1 – p. Class of discrete random variables = Binomial -- results from a binomial experiment. Conditions for a binomial experiment:

15 Examples of Binomial Random Variables A binomial random variable is defined as X=number of successes in the n trials of a binomial experiment.

16 Finding Binomial Probabilities p = probability win = 0.2; plays of game are independent. X = number of wins in three plays. What is P(X = 2)? for k = 0, 1, 2, …, n Example: Probability of Two Wins in Three Plays

17 Binomial Probability Distribution Binomial distribution is based on events in which there are only two possible outcomes on each occurrence. Example: Flip a coin 3 times the possible outcomes are (heads = hits; tails = misses): HHH, HHT, HTT, TTT, TTH, THH, THT, AND HTH

18 Binomial Probability Distribution Example: Flip a coin 3 times the possible outcomes are (call heads = hits; tails = misses): Possible Outcomes of Coin Flipped 3 times OutcomeNo. Hits (x) HHH HHT THH HTH HTT THT TTH TTT Frequency Dist of data Xf

19 Binomial Probability Distribution

20 Probability Associated with Hits HitsFrequencyProbability

21 Binomial Probability Distribution

22 Binomial Probability Distribution The preceding bar graph is symmetrical; this will always be true for the binomial distribution when p= 0.5.

23 Expected Value and Standard Deviation for a Binomial Random Variable For a binomial random variable X based on n trials and success probability p,

24 Example: Extraterrestrial Life? 50% of large population would say “yes” if asked, “Do you believe there is extraterrestrial life?” Sample of n = 100 is taken. X = number in the sample who say “yes” is approximately a binomial random variable. In repeated samples of n=100, on average 50 people would say “yes”. The amount by which that number would differ from sample to sample is about 5.