1. 7.1 Discrete and Continuous Random Variable

2.  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.

3.  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)

4.  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)

5. Lists their values and their probabilities.  Two Requirements that the probabilities must satisfy  1- p has to be between 0 and 1.  2- p₁ + p₂ + … = 1 Value of Xx₁x₂x₃.... Probabilityp₁p₂p₃....

6.  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:  P(Grade is a B or higher) = P(3 or 4)= P(3) +P(4) = 0.3 + 0.15=0.45 Grade01234 Prob.0.100.150.30 0.15

7.  used to picture the probability distributions of a discrete random variable.  Create a histogram for the above data:

8. 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)=1/4P(x=2)=3/8 P(x=3)=1/4P(x=4) = 1/16

9.  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 Number of heads 01234 Probability 1/161/43/81/41/16

10.  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.

11.  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).

12.  What is P(.3 < x <.7) =.4  P(x<.5)= 0.5  P(x >.8)= 0.2  P(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

13.  CONTINUOUS RANDOM VARIABLE X- takes all values in an interval of numbers  PROBABILITY DISTRIBUTION- density curve Difference between the two random variables: discrete (specific values) continuous (intervals)

14.  N ( μ, σ ) = N(mean, standard deviation)  The standardized variable is Z= (x-µ)/σ

15.  What is P(p.32)  Step 1: Draw and shade your normal curve:

16.  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