6.9 – Discrete Random Variables IBHLY2 - Santowski

(A) Random Variables Now we wish to combine some basic statistics with some basic probability  we are interested in the numbers that are associated with situations resulting from elements of chance i.e. in the values of random variables We also wish to know the probabilities with which these random variables take in the range of their possible values  i.e. their probability distributions

(A) Random Variables So 2 definitions need to be clarified: (i) a discrete random variable is a variable quantity which occurs randomly in a given experiment and which can assume certain, well defined values, usually integral  examples: number of bicycles sold in a week, number of defective light bulbs in a shipment discrete random variables involve a count (ii) a continuous random variable is a variable quantity which occurs randomly in a given experiment and which can assume all possible values within a specified range  examples: the heights of men in a basketball league, the volume of rainwater in a water tank in a month continuous random variables involve a measure

(B) CLASSWORK CLASSWORK: (to review the distinction between the 2 types of random variables) Math SL text, pg 710, Chap29A, Q1,2,3 Math HL Text, p 728, Chap 30A, Q1,2,3

(C) The Distributions of Random Variables For any random variable, there is an associated probability distribution: the discrete probability distribution is associated with discrete random variables  so this probability distribution describes a discrete random variable in terms of the probabilities associated with each individual value that the variable may take the normal distribution is associated with continuous random variables we will initially consider only discrete data and their associated probability distributions

(C) The Distributions of Random Variables We will toss three coins. The random variable, X, will represent the number of heads obtained. Construct a table and a graph to represent the discrete probability distribution the probability of exactly 1 head in three tosses will be written as P(X=1) (which I can read as the probability that my random variable (# of heads) has the value 1 i.e. 1 head is tossed) It can be determined in many different ways  I will use binomial probability distribution ((p + q) 3 ) from our last section  C(3,1) x (0.5) 1 x (0.5) 3-1 = 3 x 0.5 x 0.25 = Or I could use a GDC and determine binompdf(3,0.5,1) = (Or I could use the Fundamental Counting Principle ==> p(H) x p(T) x p(T) x C(3,1) ) = Likewise, I could do similar calculations to find the associated probabilities for 0,1,2,3 Heads ==> I will write this as P(X = x) and equate it to C(3,x) x (0.5) x x (0.5) 3-x, x = 0,1,2,3

(C) The Distributions of Random Variables I get the following table: I get the following graph: xP(X=x)

(C) The Distributions of Random Variables ex 2.Of the 15 light bulbs in a box, 5 are defective. Four bulbs are chosen at random from the box. Let the random variable, X, represent the number of defective bulbs selected. Construct a table and graph to represent this distribution. (NOTE: the events are NOT independent (as selecting a defective bulb first, now influences the probabilities of the selection of a second bulb  therefore, binompdf(4,1/3,x) will give us different answers than the following approach: the number of ways of selecting x defective bulbs from the 5 is C(5,x) the number of ways of selecting (4 - x) non-defective bulbs is C(10, 4-x) the number of ways of selecting 4 bulbs from 15 is C(15,4) so our basic probability formula would be (# of specific events) ) (total # of events) = [C(5,x) x C(10,4-x)] ) [C(15,4)]

(C) The Distributions of Random Variables I get the following table:I get the following graph: xP(X = x)

(D) Homework SL Math text, Chap 29B, p712, Q1-7 HL Math text, Chap 30B, p730, Q1-11