587514023-Probability-and-Statistics-Module-1.pptx

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1. Probability and Statistics Random Variables By:Mr. Jude Thaddeus Metante

2. Random Variable A V ariable is any information, attribute, characteristic, number or a quantity that describes a person, place, event, thing or idea that can be measured or counted. A variable can be : Q ualitative - a variable that is not numerical. It describes data that fits into category. Q uantitative - a numerical variables: counts, percent, numbers

3. Q ualitative (Example) Q uantitative (Example) Eye colors (variable includes: blue, green, brown) Number of pets owned (1, 2, 3 or 6) Municipality (variable includes: Consolacion , Liloan , Minglanilla ) Bank account balance (P1000, P3000 or P600) Dog Breed (variable includes: Shih Tzu, Siberian Husky, German Sheperd ) High School grade average (85, 90, 89.5 or 76) Class in College (variable includes: Junior, Senior, Freshman, Sophomore)

4. Q uantitative variable can either be discrete variable - Discrete Data can only take certain values. a quantitative variable whose value can only be attained through counting. It can be finite in number of possible values or countably infinite if the counting process has no end. b) c ontinuous - Continuous Data can take any value (within a range). A continuous variable is a quantitative variable that can assume an infinitely many, uncountable number of real number values. The value given to an observation can include values as small as the instrument of measurement allows.

5. a) discrete variable Example : the number of students in a class (We can't have half a student!) the results of rolling 2 dice ( Only has the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 we cannot get a value of 7.8) b) C ontinuous A person's height: could be any value (within the range of human heights), not just certain fixed heights, Time in a race: you could even measure it to fractions of a second, A dog's weight, The length of a leaf, Lots more!

6. A market researcher company requested all teachers of a particular school to fill up a questionnaire in relation to their product market study. The following are some of the information supplied by the teachers: • highest educational attainment • predominant hair color • body temperature • civil status • brand of laundry soap being used • total household expenditures last month in pesos • number of children in the household • number of hours standing in queue while waiting to be served by a bank teller

7. • amount spent on rice last week by the household • distance travelled by the teacher in going to school • time (in hours) consumed on Facebook on a particular day

8. 2. A survey of students in a certain school is conducted . The survey questionnaire details the information on the following variables. For each of these variables, identify whether the variable is qualitative or quantitative, and if the latter, state whether it is discrete or continuous. number of family members who are working ownership of a cell phone among family members c. length (in minutes) of longest call made on each cell phone owned per month d . ownership/rental of dwelling e . amount spent in pesos on food in one

9. e . amount spent in pesos on food in one f . occupation of household head g. total family income h. number of years of schooling of each family member i. access of family members to social media j. amount of time last week spent by each family member using the internet

10. Sample Space (S) -a list of all possible outcomes that may occur in an “ event ”. Example 3: Find the Sample Space (S) when tossing 2 coins HEADS (H) TAILS (T) , , , { } We can write S = {HH, HT, TH, TT} R andom V ariable (X) - is a variable whose value is dependent to the outcome of a well-defined random event or experiment or a set of possible values from a random experiment. (such as tossing a coin, throwing a pair of dice or drawing a card from a standard deck)

11. Sample Space HH HT TH TT X 2 1 1 Example 3: How many heads when we toss 2 coins?   How many heads means finding of Random variable (X) within a sample Space (S) From the Example 3, Sample Space Random Variables 2 Heads Or we can write X = {2, 1, 0}   Finally X = {0, 1, 2} Random Variable

12. Sample Space HH HT TH TT X 2 1 1 P robability D istribution Function P(X) - is a function that shows the relative probability that each outcome of an experiment will happen.  Example 5: From Example 4, find the probability distribution function. There are 4 sample points in the sample space         P(X) =     Therefore, Same No. of Heads = 1 therefore add:    

13. To check your answer, the total P(X) should be equal to 1 P(X) = P(X=0) + P(X=1) + P(X=2) P(X) =         Example using Die: When rolling a die there are 6 possible outcomes. Its either {1, 2, 3, 4, 5 or 6}   Therefore S= {1, 2, 3, 4, 5, 6 } DIE P(X) =         Therefore, Die Face = 6 sample points Any of these could become Random Variables Sample Space 1 2 3 4 5 6 X 1 2 3 4 5 6

14. Probability Mass Function is a probability distribution function of a discrete random variable . It assigns a probability value to each sample point . Discrete Probability Distribution is a table of values that shows the probability of any of the outcomes of an experiment.

15. Steps to Determine the Random Variables Sample Space HH HT TH TT Tossing a coin, X 2 1 1 Number of Heads Example: Tossing a coin, Rolling a die, drawing of marbles, etc.. A list of all possible outcomes that may occur in an “ event ”. Set of possible values that we want to know or we are Interested from the outcome of a well defined Experiment. Event or Experiment Sample Space Random Variable Probability Function Probability Distribution P(X) =     Therefore,             X 2 1 P(X=x) 1/4 1/2 1/4

16. Steps to Determine the Random Variables Example: Tossing a coin, Rolling a die, drawing of marbles, etc.. A list of all possible outcomes that may occur in an “ event ”. Sample Space Rolling a Die X Even numbers Set of possible values that we want to know or we are Interested from the outcome of a well defined Experiment. Event or Experiment Sample Space Random Variable Probability Function Probability Distribution

17. Example 1: Determine the random variable of the sum of the number of a pair of dice in a single throw . Die A Die B 1 2 3 4 5 6 1 [1, 1] = 2 [1, 2] = 3 [1, 3] = 4 [1,4] = 5 [1, 5] = 6 [1, 6] = 7 2 [2, 1] = 3 [2, 2] = 4 [2, 3] = 5 [2,4] = 6 [2, 5] = 7 [2, 6] = 8 3 [3, 1] = 4 [3, 2] = 5 [3, 3] = 6 [3,4] = 7 [3, 5] = 8 [3, 6] = 9 4 [4, 1] = 5 [4, 2] = 6 [4, 3] = 7 [4,4] = 8 [4, 5] = 9 [4, 6] = 10 5 [5, 1] = 6 [5, 2] = 7 [5, 3] = 8 [5,4] = 9 [5, 5] = 10 [5, 6] = 11 6 [6, 1] = 7 [6, 2] = 8 [6, 3] = 9 [6,4] = 10 [6, 5] = 11 [6, 6] = 12 Sum (x) 2 3 4 5 6 7 8 9 10 11 12 P (X=x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 DIE DIE

18. Example 1: Three coin are tossed at the same time. What are the possible outcomes? What is the probability of getting 3 heads? H T H T H T H T H T H T H T Sample Space HHH HHT HTH HTT THH THT TTH TTT x 3 2 2 1 2 1 1 P (X=x) 1 2 3 x 3 2 1 P (X=x) 1/8 3/8 3/8 1/8

19. Example 2: A coin is tossed 3 times. What are the possible outcomes? What are the random variables if we are only interested of the number of Heads? H T H T H T H T H T H T H T Sample Space x P (X=x) 1 2 3

20. Example 3: A couple is planning to have 3 children. Consider the different result that might occur in terms of gender. Determine the random variable if we are only interested in the number of boys . (ex. B B G) B G B G B G B G B G B G B G Sample Space x P (X=x) 1 2 3

21. Example 4: A dart player is trying to hit the bulls eye with each of three darts that he will throw. Each dart will either hit the bulls eye or miss the bulls eye. Determine the Sample space of the different outcomes that may Occur. Determine the random variable if we are only interested in the number of Misses. H M H M H M H M H M H M H M Sample Space x P (X=x) 1 2 3

22. Example 5: If a two child families are classified according to the sex of the first and second child. Determine the Sample space of the different outcomes. Determine the random variable if we are only interested in the number of boys. H M H M H M Sample Space x P (X=x) 1 2

23. Example 6: What are the possible outcomes when drawing two marbles from a bag containing blue, green and red marbles. Construct the probability distribution. Sample Space BB BG BR GB GG GR RB RG RR x 2 1 1 1 1 P (X=x) 1/9 4/9 4/9 B G B G 1 2 R R B G R B G R

24. THANK YOU