1. Chapter 5 Probability Distributions

2. E.g., X is the number of heads obtained in 3 tosses of a coin. [X=0] = {TTT} [X=1] = {HTT, THT, TTH} [X=2] = {HHT, HTH, THH} [X=3] = {HHH} The events corresponding to the distinct values of X are incompatible. The union of these events is the entire sample space. A random variable can be discrete or continuous. Random variables

3. Probability distribution

5. Figure 5.1 (p. 176) The probability histogram of X, the number of heads in three tosses of a coin. Probability histogram

6. Example: Two dice are tossed. The possible outcomes are shown below:

7. Probability distribution for the two dice example The event that the sum of face values is 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. Thus P(sum=7) = 6/36 Similarly, P(sum=8) = 5/36 P(sum=9) = 4/36 P(sum=10) = 3/36 P(sum=11) = 2/36 P(sum=12) = 1/36 This is the probability distribution for random variable X, the sum of face values of two dice. How can we use it in practice? See next slide. P(sum=6) = 5/36 P(sum=5) = 4/36 P(sum=4) = 3/36 P(sum=3) = 2/36 P(sum=2) = 1/36

8. Want to win Monopoly Game? Learn Counting and Probability Your opponent’s token is in one of the squares His turn consists of rolling two dice and moving the token clockwise on the board the number of squares indicated by the sum of dice values When his token lands on a property that is owned by you, you collect rent It is more advantageous to have houses or hotels on your properties because rents are much higher than for unimproved properties. You might build houses or hotels on your properties before your opponent rolls the dice Suppose you own most of the squares following (clockwise) your opponent’s token. In which square should you build houses or hotels?

9. Probability distribution for news source preference In a university, 30% of students prefer the Internet to TV for getting news. Four students are randomly selected. Let X be the number of students sampled that prefer news from the Internet. Obtain the probability distribution of X.

10. For one student, P(I)=0.3, P(T)=0.7. The observations on 4 students can be considered independent. Thus, P(TTTT) =.7 ×.7 ×.7 ×.7 =.2401 and P(X=0) =.2401 Similarly, P(TITT) =.7 ×.3 ×.7 ×.7 =.1029 P(X=1) = 4 ×.7 3 ×.3 =.4116 P(X=2) = 6 ×.7 2 ×.3 2 =.2646 P(X=3) = 4 ×.7 ×.3 3 =.0756 P(X=4) =.3 4 =.0081

11. Important distinction between a relative frequency distribution and the probability distribution: A relative frequency distribution is a sample- based entity and is therefore susceptible to variation on different occasions of sampling. The probability distribution is a stable entity that refers to the entire population. It is a theoretical structure that serves as a model for describing the variation in the population.

12. The probability distribution of X can be used to calculate the probabilities of events defined in terms of X. Ex.: For the distribution below, P[X ≤ 1] = f(0) + f(1) =.02 +.23 =.25