Finding-the-Mean-of-Discrete-Random-Variable.pptx

Statistics and Probability

1. Statistics and Probability Pauline Misty M. Panganiban Probability Distribution

2. Objectives: Illustrate and calculate the mean of discrete random variable; Interpret the mean of a discrete random variable and Solve problems involving mean of probability distributions.

3. Probability the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

4. 1. A coin is tossed. What is the probability of getting heads? P(head) =    

5. 2. What would be the probability of picking a black card at random from standard deck of 52 cards? P(black) =    

6. 3. What is the probability of rolling , on a fair dice? a. 3 P(3) =    

7. 4. What is the probability of rolling , on a fair dice? a. an even number P(even) =    

8. 5. What is the probability of rolling , on a fair dice? a. zero P(zero) =    

9.   Reminders:

10. Properties of a Probability Distribution The probability of each value of the random variable must be between or equal to 0 and 1. In symbol, we write it as 0 ≤ P(X) ≤ 1. The sum of the probabilities of all values of the random variable must be equal to 1. In symbol, we write it as ∑P(X) = 1.

12. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. Example Y 2 4 6 8 P(Y) ¼ ¼ ¼ ¼

13. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. Y 1 3 5 P(Y) 1/3 1/6 1/3 1/4 Example

14. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. Example Y 4 7 8 9 P(Y) 1/5 1/5 2/5 1/5

15. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. Example Y 3 6 9 P(Y) 0.65 -0.15 0.50

16. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. Example Y 5 10 15 20 25 P(Y) 0.12 0.34 0.23 0.16 0.15

17. Activity

18. ACTIVITY 1. The probability of each value of the random variable must be between or equal to 0 and 1. In symbol, we write it as 0 ≤ P(X) ≤ 1.

19. ACTIVITY 2. If a coin is tossed, the probability of getting tails is 1?

20. ACTIVITY 3. The sum of the probabilities of all values of the random variable must be equal to 1.

21. ACTIVITY 4. The probability of picking a red card at random from standard deck of 52 cards is .  

22. Determine whether or not the distribution represents a probability distribution. If it does not, explain why. le Y 5 10 15 20 25 P(Y) 0.12 0.34 0.23 0.16 0.15 Y 4 10 12 15 20 P(Y) 1/8 2/8 1/8 1/8 5/8

23. Activity

24. ACTIVITY The probability of each value of the random variable must be between or equal to 0 and 1. In symbol, we write it as 0 ≤ P(X) ≤ 1.

25. ACTIVITY If a coin is tossed, the probability of getting tails is 1? bluff

26. ACTIVITY The sum of the probabilities of all values of the random variable must be equal to 1.

27. ACTIVITY The probability of picking a red card at random from standard deck of 52 cards is .  

28. Statistics and Probability Pauline Misty M. Panganiban Mean of Discrete Probability Distribution

29. The Mean of a Discrete Probability Distribution To find the mean ( ) or the expected value E(x) of a discrete probability distribution, we use the following formula: Where = mean x = value of the random variable P(X) = is the probability value of the random a random variable  

30. Example 1 Find the mean of the discrete random variable X with the following probability distribution. Step 1: Construct the probability distribution X P(X) 3 1/8 2 3/8 1 3/8 1/8

31. Example 1 Step 2 : Multiply the value of the random variable X by the corresponding probability. X P(X) X • P(X) 3 1/8 3/8 2 3/8 6/8 1 3/8 3/8 1/8

32. Solution to Example 1 Step 3: Add the results obtained in Step 2. =  

33. Example 2 Surgery Patients The probabilities that a surgeon operates on 3, 4, 5, 6, or 7 patients in any day are 0.15, 0.10, 0.20, 0.25, and 0.30, respectively. Find the average number of patients that a surgeon operates on a day.

34. Solution to Example 2 Step 1: Construct the probability distribution for the random variable X representing the number of patients that a surgeon operates on a day. The probabilities that a surgeon operates on 3, 4, 5, 6, or 7 patients in any day are 0.15, 0.10, 0.20, 0.25, and 0.30, respectively. Number of Patients X Probability P(X) 3 0.15 4 0.10 5 0.20 6 0.25 7 0.30

35. Solution to Example 2 Step 2: Multiply the value of the random variable X by the corresponding probability. Number of Patients X Probability P(X) X • P(X) 3 0.15 0.45 4 0.10 0.40 5 0.20 1.00 6 0.25 1.50 7 0.30 2.10

36. Solution to Example 2 Step 3: Add the results obtained in Step 2. The average number of patients that a surgeon will operate in a day is 5.45.  

37. Summary To find the mean of the probability distribution , Construct the probability distribution for the random variable X Multiply the value of the random variable X by the corresponding probability. Add the results obtained in Step 2.

38. Illustrative Examples Find the mean of the probability distribution of the random variable X, which can take only the values 1, 2, and 3, given that P(1) = 10/33, P(2) = 11/33 , and P(3) = 12/33.

39. The probabilities of a machine manufacturing 0, 1, 2, 3, 4, or 5 defective parts in one day are 0.75, 0.17, 0.04, 0.025, 0.01, and 0.005, respectively. Find the mean of the probability distribution.

40. QUIZ: A bakeshop owner determines the number of boxes of pandesal that are delivered each day. Find the mean of the probability distribution shown. Number of Boxes X Probability P(X) 35 0.10 36 0.20 37 0.30 38 0.30 39 0.10

41. Thank you!