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7.1 Discrete and Continuous Random Variable Calculate the probability of a discrete random variable and display in a graph. Calculate the probability of a continuous random variable using a density curve. We toss a coin 4 times. Our outcome is HTTH. Let x=# of heads, therefore x=2. If we got TTTH, then x=1. The values of x are 0,1,2,3,4. (x is called a random variable) RANDOM VARIABLE: a variable whose value is a numerical outcome of a random phenomenon DISCRETE RANDOM VARIABLE X: has a countable # of outcomes (possible values) Lists their values and their probabilities. Value of X x₁ x₂ x₃ .... Probability p₁ p₂ p₃ .... Two Requirements that the probabilities must satisfy 1- p has to be between 0 and 1. 2- p₁ + p₂ + … =1 Ex. 1: The instructor of a large class gives 15% each of A’s and D’s, 30% each of B’s and C’s, and 10% F’s. Choose a student at random from this class. To “choose at random” means to give every student the same chance to be chosen. The student’s grade on a four point scale (A=4) is a random variable x: Grade 0 1 2 3 4 Prob. 0.10 0.15 0.30 0.30 0.15 P(Grade is a B or higher) = P(3 or 4)= P(3) +P(4) = 0.3 + 0.15=0.45 used to picture the probability distributions of a discrete random variable. Create a histogram for the above data: Grades 0.4 0.3 0.2 Grades 0.1 0 0 1 2 3 4 We Know: 1- outcomes ( H or T) 2- independent There are 16 possible outcomes. For ex: P(HTTH)= 1/16 What is P(x=0)= 1/16 (TTTT) P(x=1)= 1/4 P(x=3)= 1/4 P(x=2)=3/8 P(x=4) = 1/16 Number of heads 0 1 2 3 4 Probability 1/16 1/4 3/8 1/4 1/16 What is the probability of tossing at least 1 head? P(x≥1)=1-P(0)= 1- (1/16)= 15/16 What is the probability of tossing no more than 3 heads? P(x≤3)= P(x<4)= 1- P(x=4)= 1-(1/16)=15/16 A) 1% B) All probabilities are between o and 1. They also add up to 1. C) P(x≤3)= 0.94 D) P(x<3)=0.86 E) P(x≥4)=P(x>3)= 0.06 F)Let 01-48=class 1 ,49-86=class2 87-94=class 3, 95-99=class 4, 00= class 5. Use a RDT to repeatedly generate 2 digit #’s to find the proportion of those from 01-94. Look at Figure 7.4 pg. 375 (spinner) A spinner generates a random number between 0 and 1. What is the sample space? S={ 0 ≤ x < 1 } *We cannot assign probabilities to each individual value of x and then sum, because there are infinitely many possible values. -Instead we use intervals (area under a density curve)!!! (A new way of assigning probabilities directly to events). What is P(.3 < x < .7) = .4 P(x<.5)= 0.5 P(x > .8)= 0.2 P(x < .5 or x > .8)= 0.7 P(x = .8)= 0 We call X a continuous random variable because its values are not isolated #’s but an entire interval of #’s CONTINUOUS RANDOM VARIABLE Xtakes all values in an interval of numbers PROBABILITY DISTRIBUTIONdensity curve Difference between the two random variables: discrete (specific values) continuous (intervals) N ( μ , σ ) = N(mean, standard deviation) The standardized variable is Z= (x-µ)/σ What is P(p< .28 or p> .32) Step 1: Draw and shade your normal curve: Step 2: Convert to z-scores P( p< .28)= P(z< (0.28-0.3)/0.0118)= P(z<-1.69)= 0.0455 P(p > .32) =P(z> (0.32-0.3)/0.0118)= P(z>1.69)= 0.0455 0.0455+0.0455= 0.091