Notes 5.4: Discrete Random Variables



Learning Target Discrete Random VariablesRandom Variable: A variable whose value is subject to variations due to chance.Random variables are introduced as X = in words state what you are interested in (heads or seniors or….).?I can define a random variable in terms of X = (or any other capital letter)?Example 1: In a coin toss, the random variable could be the number of heads (or tails) X = number of heads in a coin tossIn a group of students the random variable could be the number of seniors (or males, or …..) S = number of seniors in a group of students.Random variables should always be defined at the beginning of your problem.There are 2 types of random variables:Discrete Random Variables: These random variables take on countable values and have a fixed amount (obtained by counting)Continuous Random Variables: These random variables take on a range of values and have an infinite amount(obtained by measuring)?I can distinguish between a discrete random variable and a continuous random variable ?Example 2: Identify and define (X = ) the type of random variable in the following situations:Number of red marbles in a jarDiscrete Random Variable; X = number of red marblesHeights of students in classContinuous Random Variable; X = heights of studentsTime is takes for students to get to school in the morningContinuous Random Variable; X = time it takes to get to schoolNumber of girls in a family of 4 childrenDiscrete Random Variable; X = number of girls (in a family of 4)Once you determine and define the type of random variable you can determine the probability of an event occurring. ?I can determine the probabilities of discrete random variables and represent these in a probability distribution table ?Since a discrete random variable has countable values, we already know how to compute the probability of each outcome. P(outcome) = number of times the outcome occurstotal possible outcomesDiscrete Random Variable outcomes and probabilities are represented in a probability distribution table.Probability Distribution Table: Lists the possible outcomes of discrete random variables and the probability of each outcome.Probability Distribution Histogram: Shows the probability of each outcome in a histogram.Example 3: Determine the probability and create a probability distribution table for the random variable X = the sum of a dice toss.Define the random variable X on the top of the tableList the possible outcomes for the sum of a dice toss in the first row X List the probabilities (number of ways the outcome occurs out of total number of outcomes) in the second row P(X). Remember the probability of a sample space = 1 (so the sum of all the P(X) = 1)X= the sum of a dice tossX23456789101112P(X)1/362/363/364/365/366/365/364/363/362/361/3641357557747000Remember in a toss of a pair of dice there are 36 outcomes (6 x 6). You can create a dice chart to help count outcomes if necessary.7372356985000Probability histogram based on the distribution table.You can use the table or histogram to answer the following questions:P(X = 4) means the probability that the sum on a dice toss is 4 P(X = 4) = 3/36P(X = 8) = 5/36P(X > 10) = P(10) or P(11) or P(12) = 3/36 + 2/36 + 1/36 = 6/36 or 1/6Example 4: Determine the probability and create a probability distribution table for the random variable X = number of girls in a family of 3 children.List the outcomes of 3 children to help you determine the probability: (8 outcomes; 2 x 2 x 2)BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGGX= number of girls in a family of 3 childrenX0123P(X)1/83/83/81/8Check sum or P(X) = 1Create a histogram, showing the probability of each outcome (based on the table above):Find the probability that there are no girls in a family with 3 children.P (X = 0) = P (X < 3) = The probability that there are 3 or less girls in a family of 3 children is 1 (100%)P (X > 3) = The probability that there are more than 3 girls in a family of 3 children is 0P ( X = 2) = The probability that there are 2 girls in a family of 3 children is 3/8P ( X > 0) = The probability that a family with 3 children any girls (more than 0 girls) is 7/8 ................
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