Statistics and Probability- Random Variables and Probability Distribution

Statistics and Probability Lesson

1. STATISTICS AND PROBABILITY RANDOM VARIABLES AND PROBABILITY DISTRIBUTION

2. STATISTICS The word Statistics actually comes from the word “state” because government have been in the statistical activities especially the conduct of censuses. Is defined as a science that studies data to be able to make a decision. It involves the methods of collecting, processing, summarizing and analyzing data in order to provide answers or solutions to an inquiry.

3. PROBABILITY Probability is simply how likely something is to happen Whenever were unsure about the outcome of an event, we can talk about the probabilities of certain outcomes---how likely they are. The analysis of events governed by probability is called statistics.

4. ACTIVITY Activity is any activity which can be done repeatedly under similar conditions Examples: answering multiple choice question answering true or false question tossing a coin

5. SAMPLE SPACE The set of all POSSIBLE outcomes of an experiment Examples: answering multiple choice question, possible outcome: a, b, c, d answering true or false question; possible outcome: true, false tossing a coin, possible outcome: TAIL, HEAD

6. UNIVERSE is the collection or set of units or entities from whom we got the data. VARIABLE is a characteristics that is observable or measurable in every unit of the universe POPULATION is the set of all possible values of a variable. SAMPLE is the subgroup of a universe or of a population.

7. Lesson 1: RANDOM VARIABLES It is a function that associates a real number to each element in the sample space. It is a result of chance in an event that you can measure or count. It is a numerical quantity that is assigned to the outcome of an experiment. NOTE: We use capital letters to represent a random variable

8. Steps in Finding RANDOM VARIABLE Step 1 : List the sample space, S Step 2 : Count the number of the assigned Value in each outcome and assign this number to this outcome. Step 3 : Make a Conclusion

9. RANDOM VARIABLES EXAMPLE Example #1 Suppose two coins are tossed and we are interested to determine the number of tails that will come out. Let us use T to represent the number of tails that will come out. Determine the values of the random variable T.

10. Solution to Example #1 Step 1 : List the sample space, S S = {HH, HT, TH, TT)

11. Solution to Example #1 Step 2 : Count the number of the assigned Value in each outcome and assign this number to this outcome. OUTCOME NUMBER OF TAILS (Value of T) HH HT 1 TH 1 TT 2

12. Solution to Example #1 Step 3: Make a conclusion . The values of the random variable T (number of tails) in this example are 0, 1, and 2

13. Solution to Example #1 Step 3: Make a conclusion . The values of the random variable T (number of tails) in this example are 0, 1, and 2

14. RANDOM VARIABLES EXAMPLE Example #2 Two balls are drawn in succession without replacement from an urn containing 5 orange balls and 6 violet balls. Let V be the random variable representing the bnumber of violet balls. Find the values of the random variable

15. Solution to Example #2 Step 1 : List the sample space, S S = {OO, OV, VO, VV)

16. Solution to Example #1 Step 2 : Count the number of the assigned Value in each outcome and assign this number to this outcome. OUTCOME NUMBER OF VIOLET BALLS (Value of V) OO OV 1 VO 1 VV 2

17. Solution to Example #1 Step 3: Make a conclusion . The values of the random variable V (number of violet balls) in this example are 0, 1, and 2

18. Lesson 2: DISCRETE AND CONTINUOUS RANDOM VARIABLE A random variable may be classified as discrete and continuous. Discrete variable has a countable number of possible values. Continuous random variable can assume an infinite number of values in one or more intervals..

19. DISCRETE RANDOM VARIABLE # OF PENS IN A BOX # OF ANTS IN ACOLONY # OF RIPE BANANAS IN A BASKET # OF COVID-19 POSITIVE CASES IN BAGAC # OF DEFECTIVE BATTERIES POSSIBLE VALUES ARE COUNTABLE, WHOLE NUMBER AND A FIX VALUE

20. CONTINUOUS RANDOM VARIABLE AMOUNT OF ANTOBIOTIC IN VIAL LENGTH ELECTRIC WIRES VOLTAGE OF CAR BATTERIES WEIGHT OF NEWBORN IN THE HOSPITAL AMOUNT OF SUGAR IN A CUP OF COFFEE POSSIBLE VALUES ARE NOT COUNTABLE , NOT WHOLE NUMBER AND DOES NOT HAVE A FIX VALUE

21. Lesson 3: PROBABILITY DISTRIBUTION AND HISTOGRAM FOR THE PROBABILITY DISTRIBUTION Steps in Constructing Probability Distribution and Histogram for the Probability Distribution Step 4 : Construct the frequency distribution of the values of the random variable T.

22. Steps in Constructing Probability Distribution and Histogram for the Probability Distribution Step 5 : Construct the probability distribution of the random variable T by getting the probability of occurrence of each value of the random variable. Step 6 : Construct the probability histogram.

23. *We use example 1 on RANDOM VARIABLES Number of Tails (Value of T) Number of Occurrence (Frequency) 1 1 2 2 1 Total 4 Step 4:

24. *We use example 1 on RANDOM VARIABLES Step 5: Number of Tails (Value of T) Number of Occurrence (Frequency) Probability P(T) 1 ¼ 1 2 2/4 or 1/2 2 1 ¼ Total 4 4/4 = 1 T 2 1 P(T) 1/4 1/2 1/4

25. Step 6: Construct Histogram

26. MEAN, VARIANCE AND STANDARD DEVIATION