1. Chapter 4 Discrete Random Variables

2. Two Types of Random Variables Random Variable –Variable that assumes numerical values associated with random outcomes of an experiment –Only one numerical value is assigned to each sample point Two types of Random Variable –Discrete –Continuous

3. Two Types of Random Variables Discrete Random Variable –Random variable that has a finite, or countable number of distinct possible values –Example – number of people born in July Continuous Random Variable –Random variable that has an infinite number of distinct possible values –Average age of people born in July

4. Probability Distributions for Discrete Random Variables Two requirements that must be satisfied: for all values of x Where the summation of p(x) is over all possible values of x 1. 2.

5. Probability Distributions for Discrete Random Variables Experiment - tossing 2 coins simultaneously Random variable X – number of heads observed X can assume values of 0, 1 and 2 Calculate the probability associated with each value of X Probability Distribution for Coin-Toss Experiment Xp(x) 0¼ 1½ 2¼

6. Probability Distributions for Discrete Random Variables Probability Distribution of Discrete Random Variable X – Other forms

7. Expected Values of Discrete Random Variables The mean, or expected value of a discrete random variable is: Expected Value of x (number of heads observed) XP(x)Xp(x) 0¼0 1½½ 2¼½ Expected Value1

8. Expected Values of Discrete Random Variables The variance of a discrete random variable is: and standard deviation is

9. Expected Values of Discrete Random Variables Probability Rules for a Discrete Random Variable Chebyshev’s RuleEmpirical Rule Applies to any distribution Applies to mound- shaped and symmetric distributions

10. The Binomial Random Variable Binomial Random variable –An experiment of n identical trials –2 possible outcomes on each trial, denoted as S (success) and F (failure) –Probability of success (p) is constant from trial to trial. Probability of failure (q) is 1-p –Trials are independent –Binomial random variable – number of S’s in n trials

11. The Binomial Random Variable Computer retailer selling desktop (D) and laptop (L) PCs online. Sales of 80% desktop, 20% laptop. What is the probability that next 4 sales are Laptops? Sample points for next 4 online purchases DDDDLDDD DLDD DDLD DDDL LLDD LDLD LDDL DLLD DLDL DDLL DLLL LDLL LLDL LLLD LLLL

12. The Binomial Random Variable Use multiplicative rule to calculate probabilities of the possible outcomes P(DDDD) =.8*.8*.8*.8=.8 4 =.4096 P(LDDD) =.2*.8*.8*.8=.2*.8 3 =.1024 ….. P(LLLL) =.2*.2*.2*.2=.2 4=.0016

13. The Binomial Random Variable What is the probability that 3 of the next 4 online sales are laptops? P(3 of the next 4 customers purchase laptops) = 4(.2) 3 (.8)=4(.0064) =.0256 What is the probability that 3 of the next 4 online sales are desktops? P(3 of the next 4 customers purchase desktops) = 4(.8) 3 (.2)=4(.1024) =.4096 Do you see a pattern?

14. The Binomial Random Variable Formula for the probability distribution p(x) Where p = probability of success on single trial q = 1-p n = Number of trials x = number of successes in n trials

15. The Binomial Random Variable P(3 of the next 4 customers purchase laptops) = 4(.2) 3 (.8)=4(.0064) =.0256 x=3, n=4

16. The Binomial Random Variable P(3 of the next 4 customers purchase desktops) = 4(.8) 3 (.2)=4(.1024) =.4096 x=3, n=4

17. The Binomial Random Variable Mean: Variance: Standard deviation

18. The Binomial Random Variable Using Binomial Tables Binomial tables are cumulative tables, entries represent cumulative binomial probabilities Make use of additive and complementary properties to calculate probabilities of individual x’s, or x being greater than a particular value.

19. The Binomial Random Variable If x < 2, and p =.2, n =10, then P(x<2) =.678 If x = 2, and p =.2, n =10, then P(x=2) = P(x<2) - P(x<1)=.678-.376 =.302 If x >2, and p =.2, n =10, then P(x>2) = 1- P(x<2) =1-.678 =.322.30.20.10.05.01.383.678.930.9881.0002.149.376.736.914.9961.650.879.987.9991.0003 4 0.850.967.9981.000.028.107.349.599.904 Binomial probabilities for n=10 (partial table) p k