MODULE 1: Random Variables and Probability Distributions Quarter 3 Statistics and Probability .pptx

This contains the basic concepts of Statistics and Probability like Random Variables and
Probability Distributions
since this is the first topic in Statistics and Probability

1. 3rd Quarter Statistics and Probability

2. Module 1: Random Variables and Probability Distributions

3. OBJECTIVES: At the end of the lesson, the students are expected to: a. illustrate a random variable (discrete and continuous ); b. distinguish between a discrete and a continuous random; c. find the possible values of a random variable.

4. Let Us Try Direction: Classify each random variable as discrete or continuous .

5. 1. The number of arrivals at an emergency room between midnight and 6:00a.m. 2. The weight of a sack of rice labeled “50 kilograms.”

6. 3. The duration of the next outgoing telephone call from a business office. 4. The number of kernels of popcorn in a five-kilo container.

7. 5. The number of applicants for a job.

8. variable is a placeholder for real number values that can be assigned to it LESSON 1: RANDOM VARIABES Some examples of variables include X = number of heads or Y = number of cell phones or Z = running time to movies.

9. LESSON 1: RANDOM VARIABES E ngrossed in some numerals associated with the outcomes. For example, if a coin is tossed twice, the set of all possible outcomes (S) of the experiment is: S = {TT, TH, HT, HH}

10. LESSON 1: RANDOM VARIABES S = {TT, TH, HT, HH} Sample space (S) = is a collection or a set of possible outcomes of a random experiment. The subset of possible outcomes of an experiment is called events.

11. LESSON 1: RANDOM VARIABES A sample space may contain several outcomes which depends on the experiment . If it contains a finite number of outcomes , then it is known as discrete or finite sample spaces .

12. LESSON 1: RANDOM VARIABES X = {0, 1, 2}. Then X is called a random variable.

13. RANDOM VARIABES 01.

14. RANDOM VARIABES A random variable is a variable whose value is determined by the outcome of a random experiment. (upper case) X for the random variable and lower case x1, x2, x3,... for the values

15. RANDOM VARIABES U pper case letter for the random variable and lower case letter x1 , x2, x3,... for the values

16. Types of Random Variables

17. D iscrete random variable said to be a random variable X and it has a finite number of elements or infinite but can be represented by whole numbers . These values usually arise from counts

18. Finite sets or countable sets : sets having a finite/countable number of members. Examples of finite sets: P = {0, 3, 6, 9, …, 99} Q = {a: a is an integer, 1 < a < 10} A set of all English Alphabets (because it is countable).

19. Infinite sets or uncountable sets: the number of elements in that set is not countable and it cannot be represented in a Roster form. Examples of infinite sets: A set of all whole numbers, W = {0, 1, 2, 3, 4,…} A set of all points on a line The set of all integers

20. Continuous random variable said to be a random variable Y and it has an infinite (unaccountable) number of elements and cannot be represented by whole numbers. These values usually come from measurements.

21. PRACTICE A teacher’s record has the following: (a) scores of students in a 50-item test , (b) gender, ( c) height of the students. Classify each whether discrete or continuous variable.

22. ANSWERS Scores of students in a 50-item test are a discrete random variable . Gender is also a discrete random variable. Height of the students is regarded as a continuous random variable.

23. PRACTICE MORE DIRECTION: Give the set of possible values for each random variable. 1. The number of coins that match when three coins are tossed at once. 2. The number of games in the next World Cup Series (best of four up to seven games). 3. The amount of liquid in a 12-ounce can of soft drink. 4. The average weight of newborn babies born in the Philippines in a month. 5. The number of heads in two tosses of coin.

24. ASSIGNMENT Write the possible values of each random variable. Use a ¼ sheet of paper (YELLOW PAPER). a. X = number of heads in tossing a coin thrice b. Y = dropout rate (%) in a certain high school

25. Let Us Practice More DIRECTION: Match the following with each letter on the probability line.

26. Let Us Practice More 1 . A male student is chosen in a group of 4 where 1 is female.

27. Let Us Practice More ____ 2. If you flip a coin, it will come down heads.

28. Let Us Practice More ____ 2. If you flip a coin, it will come down heads.

29. Let Us Practice More ____ 3. It will be daylight in Davao City at midnight.

30. Let Us Practice More ____ 4. Of the 40 seedlings, only 10 survived.

31. Let Us Practice More ____ 5. The third person to knock on the door will be a female.

32. Discrete Probability Distribution is a table listing all possible values that a discrete variable can take on, together with the associated probabilities. LESSON 2: PROBABILITYDISTRIBUTIONS

33. The function f(x) is called a Probability Density Function for the continuous random variable X where the total area under the curve bounded by the x-axis is equal 1. LESSON 2: PROBABILITY DISTRIBUTIONS

34. Relative frequency of an event is the number of times the event occurs divided by the total number of trials . For instance, if you observed 100 passing cars and found that 23 of them were red, the relative frequency would be 23/100. PROBABILITIES AS RELATIVE FREQUENCY

35. EXAMPLE: Flip 3 coins at same time. Let random variable x be the number of heads showing. PROBABILITIES AS RELATIVE FREQUENCY

36. EXAMPLE: Make a probability distribution for X, sum of 2 rolled dice. PROBABILITIES AS RELATIVE FREQUENCY

37. Example 1 The weight of a jar of coffee selected is a continuous random variable. The following table gives the weights in kg of 100 jars of coffee recently filled by the machine. It lists the observed values of the continuous random variable and their corresponding frequencies.

38. Find the probabilities for each weight category?

39. Let X represent a discrete random variable with the probability distribution function P(X). Then the expected value of X denoted by E(X) , or μ , is defined as: E(X ) = μ = Σ (xi × P(xi)) EXPECTED VALUE OF A RANDOM VARIABLE

40. To calculate this, we multiply each possible value of the variable by its probability , then add the results . Σ (xi × P(xi)) = {x1 × P(x1)} + {x2 × P(x2)} + {x3 × P(x3)} + ... E(X) is also called the mean of the probability distribution. EXPECTED VALUE OF A RANDOM VARIABLE

41. To calculate this, we multiply each possible value of the variable by its probability , then add the results . Σ (xi × P(xi)) = {x1 × P(x1)} + {x2 × P(x2)} + {x3 × P(x3)} + ... E(X) is also called the mean of the probability distribution. EXPECTED VALUE OF A RANDOM VARIABLE

42. Complete the table and find the expected value of x ?

43. Complete the table and find the expected value of x ?

44. Let X represent a discrete random variable with probability distribution P(X). The variance of X denoted by V(X) or σ^2 is defined as : V(X) = Σ[{X − E(X )} ^ 2 × P(X) ] VARIANCE OF A RANDON VARIABLE

45. Let X represent a discrete random variable with probability distribution P(X). The variance of X denoted by V(X) or σ^2 is defined as : V(X) = Σ[{X − E(X )} ^ 2 × P(X) ] VARIANCE OF A RANDON VARIABLE

46. Complete the table and find variance?