1. 14/6/1435 lecture 10 Lecture 9

2. The probability distribution for the discrete variable Satify the following conditions P(x)>= 0 for all x

3. Probability Functions and Probability Distributions Example 1: In an experiment of tossing a fair coin three times observing the number of heads (X) find: 1. The probability distribution table 2. The mathematical expectation ( mean) 3. Construct the probability histogram Jan 2009 3

4. S = { HHH, HHT, HTH, THH, TTT, THT, TTH, HTT } At any point x, the number of heads are S = { HHH, HHT, HTH, THH, TTT, THT, TTH, HTT } 23 0 2 2 1 1 1 From the sample space

5. 1.The probability distribution table قيمة X 0123 P(X=X)1/83/8 1/8

6. عدد ظهور الوجه H 0123 الاحتمال 1/8 3/8 1/8 Mathematical Expectation 2.The mathematical expectation ( mean)

7. Probability Histogram حp( x) 3 / 8 2/ 8 1/ 8 0 1 2 3 3. probability histogram

8. . Example 2: In an experiment of tossing a fair coin three times observing the absolute difference between the number of H and T find: 1. The probability distribution table 2. The expectation 3. The mean 4. Construct the probability histogram Jan 2009 8

9. S = { HHH, HHT, HTH, THH, TTT, THT, TTH, HTT } 13 3 1 1 1 1 1 Its range is a random variable defined in Y = {1,3} The absolute difference and the range

10. Jan 2009 10 قيمة Y 13 P(Y=y) 3/4 1/4 1.The probability distribution table What is the probability that Y= 1

11. Mathematical Expectation مثا ل قيمة Y 13 المجموع P(y)3/41/41 Y p(y)3/4 6/4

12. In an experiment of rolling two fair dice, X is defined as the sum of two up faces مثا ل654321 (6,1)(5,1)(4,1)(3,1)(2,1)(1,1)1 (6,2)(5,2)(4,2)(3,2)(2,2)(1,2)2 (6,3)(5,3)(4,3)(3,3)(2,3)(1,3)3 (6,4)(5,4)(4,4)(3,4)(2,4)(1,4)4 (6,5)(5,5)(4,5)(3,5)(2,5)(1,5)5 (6,6)(5,6)(4,6)(3,6)(2,6)(1,6)6 Elements of the sample space = 6 2 = 36 elements X is a random variable defined in S The range of it is {2,3,4,……….,11,12}

13. What is the probability that X= 4 i.e what is the probability that the sum of the two upper faces =4