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Mean and Variance of a Discrete Random Variable: A Comprehensive Guide with Examples, Cheat Sheet of Mathematics

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STATISTICS AND
PROBABILITY
Quarter 3 – Module 3
Mean and Variance of a
Discrete Random Variable
What I Need to Know
This module was designed and written with you in mind. It is here to indulge you on
computing the mean and variance of a discrete random variable. The scope of this
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Partial preview of the text

Download Mean and Variance of a Discrete Random Variable: A Comprehensive Guide with Examples and more Cheat Sheet Mathematics in PDF only on Docsity!

STATISTICS AND

PROBABILITY

Quarter 3 – Module 3

Mean and Variance of a

Discrete Random Variable

What I Need to Know

This module was designed and written with you in mind. It is here to indulge you on computing the mean and variance of a discrete random variable. The scope of this

module permits it to be used in many different learning situations wherein will help you analyze problems involving real-life situations. The arrangement of the lessons follows the standard sequence of the course. But the pacing in which you read and answer this module is dependent on your ability. This module is aligned with the following most essential learning competencies: 1.calculate the mean and variance of a discrete random variable (M11/12SP-IIIb-2); 2.interpret the mean and the variance of a discrete random variable (M11/12SP-IIIb-3); 3.solve problems involving mean and variance of probability distributions (M11/12SP- IIIb-4). Primarily, the scope of this module is to develop your skill in computing for the mean and variance of a discrete random variable. While going through this module, you are expected to: 1.apply the important concepts of mean and variance of a discrete random variable; 2.compute and interpret the mean and variance of a discrete random variable; and 3.solve real-life problems involving mean and variance of probability distributions.

What I Know

Let us determine how much you already know about computing the mean and variance of a discrete random variable. If you answer all the test items correctly in this pretest, then you may skip studying this learning material and proceed to the next learning module. Direction : Read and understand each item, then choose the letter of the best answer and write it on a separate sheet of paper.

  1. Which of the following are TRUE statements? I. The standard deviation of a random variable is equal to zero if it takes a single value. II. The standard deviation of a random variable can never be negative. III. As the number of observations increases, the mean of a random variable will get closer and closer to a particular value. A) I and II C) II and III B) I and III D) I, II and III
  2. Can the function (𝑥) = x + 5 21 , for x = 1, 2, and 3, be the probability distribution for some random variable? A)Yes. B) No, since probabilities can never be negative. C)No, since probabilities can never be greater than 1. D)No, since probabilities do not add up to 1.
  3. If (𝑋) =
X

, what are the possible values of X for it to be a probability distribution? A) 0, 2, 3 B) 1, 2, 3 C) 2, 3, 4 D) 3, 4, 5 For items 4-5 , consider the following discrete probability distribution: X 0 1 2 3 4 P(X) 1/5 1/5 1/5 1/5 1/ What is the expected value of the probability distribution? A) 1 B) 2 C) 3 D) 4

  1. What is the variance of the probability distribution? A) 1 B) 2 C) 3 D) 4 For items 6-9 , refer to the scenario and table provided below.

A) One of the outcomes will take place 40% of the time. B) Two of the outcomes will never happen. C) The total of the probabilities is 1. D) The probabilities range from 0 and 1. Calculating and Interpreting the Mean and Variance of a Discrete Random Variable

Lesson

1 When we make decisions in real-life scenarios, we consider all potential outcomes, their probabilities and assess the results. For example, a Grade 11 learner like you can compute for the probability that 7 or 8 ballpens and 6 or 7 yellow pads will be used in answering modules for the different subjects. A Grade 11 teacher might be able to calculate the probabilities for giving more than 4 performance tasks each quarter. These events can be computed using the mean, variance and standard deviation as soon as the likelihoods are distributed. In this lesson, you will learn how to calculate and interpret the mean and variance of a discrete random variable. The measures are, however, different from the mean and variance for sample data.

What’s In Let’s have a review first.

In a certain school, the number of learners present during an online class in Statistics per week is a random variable represented by X. The probability distribution for X is presented below.

32 34 36 38 40 Number of Learners Present in a Week (X)

  1. Find the probability that in a given week: a. at most 36 learners are present during an online class in Statistics? b. at least 36 learners are present during an online class in Statistics? P(X)

c. at least 34 learner are present during an online class in Statistics? d. exactly 40 learners are present during an online class in Statistics?

  1. What is the average number of learners present during an online class in a week?
  2. Determine the variance and standard deviation of the given random variable.

What’s New

At this point of the module, you are about to learn the concept in computing mean and variance of a discrete random variable. To understand better the concept, you will have to perform the simple activity below. Read and analyze the situation given below. Mr. Tamondong, a PE teacher, manages to give an assessment composed of 4 performance tasks in the 3rd^ Quarter of the school year. He noticed that some learners did not submit some of the performance tasks. He used to check the probability distribution of the submitted performance tasks and the data is presented below.

0 1 2 3 4 Number of Performance Tasks Submitted (PE) Based on the data offered, answer the following questions:

  1. What is the sum of all the probabilities of the given random variable (i.e. submission of performance tasks by the learners)? _____________________
  2. What have you noticed about the probability distribution? Can it take a negative value? State your reason.
  1. What is the average number of performance tasks submitted to Mr. Tamondong?

Probability Distribution (Submission)

So, the expected value or mean of the given discrete random variable is 2.40. Now that you know how to calculate the mean of a discrete random variable, you must remember that for a set of data, you also need other measures like measures of variability (i.e. variance and standard deviation) to see the difference of the value of X from the mean. Variance and Standard Deviation of a Random Variable Recall that a variance of a random variable is given by

𝟐

𝟐

𝟐

The variance is a parameter that is used to assess the spread of the distribution (i.e. how far the values of X are from the mean). Bear in mind that the greater the variance, the more the X values deviate from the mean. Alternatively, the standard deviation of a random variable is square root of the

variance. It is denoted by 𝝈𝑿 and computed using the following formula:

To demonstrate how to compute the variance and standard deviation of a discrete random variable, we will use the previous data on the number of performance tasks in PE submitted to Mr. Tamondong. For the variance, each value of X will be subtracted to the expected value (μ). Then take the square and multiply it by its probability. Afterwards, add the products. X P(X) (X)●P(X) (X - μ))^2 (X - μ))^2 ● P(X) 0 0.10 0.00 (0-2.40)^2 = 5.76 0. 1 0.10 0.10 (1-2.40)^2 = 1.96 0. 2 0.25 0.50 (2-2.40)^2 = 0.16 0. 3 0.40 1.20 (3-2.40)^2 = 0.36 0. 4 0.15 0.60 (4-2.40)^2 = 2.56 0. 𝝁𝑿 = ∑ 𝑿 ∙ 𝑷(𝑿) (^) 2.40 𝝈𝟐𝑿 = ∑(𝑿 − 𝝁)𝟐^ ∙ 𝑷(𝑿)

For the standard deviation, take the square root of the variance. 𝝈𝑿. 𝟏𝟔 Let us take another example. According to the National ICT Household Survey (NICTHS) 2019, 12.3% or 154, of the households in the Ilocos Region have used mobile phones. Suppose you know the probabilities for the number of mobile phones in use per household. Based on the prepared discrete probability distribution of the given random variable, determine the mean, variance, and standard deviation. Number of Mobile Phones in Use per Household (X)

Probability Distribution P(X) 0.87 0.04 0.06 0. Applying the concepts of the mean, variance and standard deviation in the above situation, we will obtain the following

The expected value or mean is 0.25. The variance is 0.4875. The standard deviation is.

To find for the mean, ________________________________________________________. Meanwhile, to compute for the variance, _____________________________________. In addition, to determine the standard deviation, _____________________________.

What I Can Do

Activity 4. Investigation Time!

  1. Assume that Raymund Gregory, a Grade 11 student, is scheduled to take a 20-item summative assessment in Physical Science, Statistics and Probability, and Research in Daily Life I. He knows he has a 50% chance of having a perfect score based on his previous summative evaluations in each subject. Given that X represents the number of perfect scores Raymund Gregory will obtain, determine the values and corresponding probabilities of the distribution of X.
  2. Assume that a fair coin is tossed and a fair die is rolled. In a game created by Yuno and Asta, the highest number in each case determines the winner. Asta flips a coin with the number 9 on one hand and the number 19 on the other. Yuno throws a die with tens on four sides and twenty on two sides. a. In the given situation, which character would you prefer - Yuno or Asta? Give a reason for your decision.
  3. To make the game equal, if Yuno receives P 50.00 for each win, what should Asta receive for each win?

Assessment

Let us determine how much you have learned from this module. Read and understand each item, then choose the letter of your answer and write it on your answer sheet.

  1. If 𝐹(𝑋) = 𝑋^ , what are the possible values of X for it to be a probability 150 distribution? A) 10, 30, 50 C) 50, 60, 70 B) 30, 50, 70 D) 60, 70, 80
  2. For x = 10, 20, and 30, can the function ℎ(𝑥) =
X + 5

be the probability distribution for some random variable? A) Yes. B) No, since probabilities can never be negative. C) No, since probabilities can never be greater than 1. D) No, since probabilities do not add up to 1. For items 3-5 , consider the following discrete probability distribution: X 0 1 2 3 4 P(X) 3/15 3/15 3/15 3/15 3/

  1. What is the expected value of the probability distribution? A) 1 B) 2 C) 3 D) 4
  2. What is the variance of the probability distribution? A) 1 B) 2 C) 3 D) 4
  3. What is the standard deviation of the probability distribution? A) 1.40 B) 1.41 C) 1.42 D) 1. For items 6-8 , refer to the scenario and table provided below. The number of female individuals living in the household on a randomly selected barangay is described by the following probability distribution. X 0 1 2 3 4 P(X) 0.10 0.20 0.40 0.20 0.
  4. What is the mean of the probability distribution? A) 1.0 B) 1.5 C) 2.0 D) 2.
  5. What is the variance of the probability distribution? A) 1.10 B) 1.15 C) 1.20 D) 1. 11
  1. What is the standard deviation of the probability distribution? A) 1.10 B) 1.20 C) 1.30 D) 1.
  2. If the variance of a random probability distribution is 1.05, what is the standard deviation? A) 1.0125 B) 1.1025 C) 1.125 D) 1. For questions 10-13 , refer to the following. Suppose Hananiah Hermarie, a Grade 11 learner, recorded the probability distribution for the number of yellow pads that she used in answering his modules in different subjects. X 3 4 5 6 7 P(X) 0.10 0.20 0.25 0.20 0.
  3. What is the mathematical expectation of the given probability distribution? A) 3.05 B) 3.50 C) 5.03 D) 5.
  4. What value corresponds to the variance of the distribution? A) 1.17 B) 1.71 C) 2.17 D) 2.
  5. What is the probability that Hananiah Hermarie used at least 5 yellow pads? A) 0.30 B) 0.50 C) 0.70 D) 0.
  6. Which of the following corresponds to the standard deviation of the distribution? A) 1.31 B) 1.33 C) 1.35 D) 1.
  7. Which of the following refers to the sum of the product of X and P(X) of a discrete random variable? A) variance B) probability C) expected value D) deviation
  8. Which of the following is a TRUE statement? A) The probability distribution of a discrete random variable can be negative. B) The expected value of a probability distribution is always equal to 1. C) The variance is equal to the standard deviation of a probability distribution. D) The mathematical expectation is the same with the mean value of a probability distribution of a discrete random variable.

Additional Activities

  1. Suppose John Paul Gabriel, a Grade 11 TVL HE learner, recorded the number of cookies baked per day as part of his subject Bread and Pastry Production. The data are as follows: X (no. of cookies) 15 20 25 30 35 40 P(X) 0.10 0.15 0.25 0.25 0.15 0. a. What is the probability that John Paul Gabriel will bake more than 25 cookies? b. What is the probability that John Paul Gabriel will bake less than 30 cookies? c. Find the mathematical expectation (mean) and variance of the given discrete random variable.
  2. The Grade 11 HUMSS learners wanted to know the number of ballpens used by Grade 11 learners in answering their modules in different subjects for the First Semester of the School Year 2020-2021. The following distribution was documented: X (no. of ballpens) 4 5 6 7 8 9 P(X) 0.05 0.13 0.17 0.28 0.22 0. a. What is the probability that a Grade 11 learner will use less than 7 ballpens in answering modules? 12

14

b. Variance (using the formula 𝝈𝟐𝑿 = ∑ [𝑿𝟐^ ∙ 𝑷(𝑿)] − 𝝁𝟐^ ) X P(X) X^2 X^2 ● P(X) 14 0.20 196 39. 15 0.30 225 67. 16 0.30 256 76. 17 0.20 289 57. ∑[𝑿𝟐^ ∙ 𝑷(𝑿)] = 241. 𝝈𝟐𝑿 = ∑[𝑿𝟐^ ∙ 𝑷(𝑿)] − 𝝁𝟐 𝝈𝟐𝑿 = 241.30 – (15.50)^2 𝝈𝟐𝑿 = 241.30 – 240. 𝝈𝟐𝑿 = 1. c. Standard deviation

𝝈𝑿 = √∑[𝑿

𝟐

∙ 𝑷(𝑿)] − 𝝁

𝟐

Therefore, the mean of the given discrete random variable is 15.50; the variance is 1.05 and the standard deviation is 1.02.

What’s New

At this point of the module, you are about to learn how to solve problems involving mean and variance of a discrete random variable. To understand better the concept, you will need to complete the simple activity below. Activity 1: Pick-A-Box The mother of Raymund Gregory wants him to choose one of the three boxes and pick a bill after. The three boxes containing different denominations are shown below. Images source: https://en.wikipedia.org/wiki/Banknotes_of_the_Philippine_peso Questions:

  1. What is the expected winnings for each box?
  2. Which box would you recommend to Raymund Gregory if you were his friend? Give a reason for your decision. Activity 2: Win-Win Investment Suppose your parents wanted to invest their money in a certain financial institution. The three financial institutions have the following offers: GINTO Investment PILAK Investment TANSO Investment Makisig Box Mayumi Box Maharlika Box 17 pcs. 3 pcs. 16 pcs 4 pcs 15 pcs 5 pcs

c. For Maharlika Box: X 100 200 P(X) 15/20 5/ (𝑋) = (100) + (200) = 75 + 50 = 125

  1. In comparison to the other boxes, I would advise Raymund Gregory to pick Makisig Box because it has the highest expected value. Win-Win Investment Activity
  2. The expected value of each investment is given below: GINTO (G) Investment
PILAK (P)

Investment

TANSO (T)

Investment Outcome Gain Loss Gain Loss Gain Loss Probability 0.25 0.75 0.50 0.50 0.85 0. Random variable

E(G) = 0.25(500,000) + 0.75(-50,000) = 125,000 - 37,500 = P87,500.
E(P) = 0.50(250,000) + 0.50(-80,000) = 125,000 – 40,000 = P85,000.
E(T) = 0.85(100,000) + 0.15(-5,000) = 85,000 – 750 = P84,250.

My parents should choose the offer of GINTO Investment because the expected value of P87,500 is the highest among the three investments.

  1. Despite the fact that GINTO and PILAK investments have higher expected values than TANSO investment, my parents should opt for TANSO investment because it has a better chance of preventing bankruptcy.

What’s More

Now, it’s your turn to apply the concepts on how to calculate and interpret mathematical expectation or mean of a discrete random variable. Activity 3. Solving EXPECTEDly! Direction: Read and analyze the situations. Answer the questions as required.

  1. Suppose Mahika Kagamitan Center will hold its annual raffle bonanza. A flat screen TV worth P17,500 is up for grabs with 5,000 tickets priced at P50.00 each. What is the expected value of Juan Emilio’s gain if he buys 15 tickets?
  2. Assume there are 100 prizes of P100, 50 prizes of P200, and five prizes of P1000 in a lottery. What is a fair price to charge for a ticket if there are 5,000 tickets to be distributed and sold?
  3. Assume there are three prizes available in an online raffle for a good cause: one P5,000 prize, one P3,000 prize, and one P1,500 prize. A total of 1,000 tickets will be sold for P20 each. What is the mathematical expectation if a person buys one ticket?

What I Have Learned

This is amazing! You have reached this part of the module. To ensure you have a complete understanding on the concepts relating to the calculation

of the expected value or mean and variance of a discrete random variable, it is necessary that you complete each statement below.

  1. The ____________________________ is the mean of a random variable.
  2. If expected value is positive, then a ______________ is expected.
  3. If expected value is negative, then a ______________ is likely to occur.
  4. To compute for the expected value, the formula to be used is _______________.
  5. In obtaining the mathematical expectation of a certain random variable, get the sum of all the products formed by the _____________ and ______________.

What I Can Do

Activity 4. You Can Do It! Suppose you will be designing a game or problem using expected value. By filling out the table below, you will construct a problem where the expected value represents a gain or win and/or illustrates a drawback eventually. The first item has already been presented for your reference. Gain Loss Lottery Assume^ you^ will^ choose^ a number from 1 to 100 and pay P5.00. If your number is called, you will be awarded P1000. You have 0.01 chance to win. Determine the expected value of your profit. Assume you'll choose a number from 1 to 100 and pay P15.00. You will be given P1000 if your number is called. You have a 0.01 chance of winning. Calculate your profit's expected value. Raffle Draw Insurance/Investment

Assessment

Let us determine how much you have learned from this module. Read and understand each item. Choose the letter of your answer and write it on your answer sheet.

  1. Suppose you are one of the 10,000 people who send in their name through text in an online raffle promo with 10 prizes of P5,000, 25 prizes of P2, and 50 prizes of P1,000. What will be your expected winning? A) 5 B) 10 C) 15 D) 20
  2. Your mother wants you to choose one of the three boxes and pick a bill after. The first box has two P1000-bill and eighteen P200-bill. On the other hand, the second box contains ten P1000-bill and forty P100-bill and the third box has 14 P20-bill. Which of the following claims is true with regards to the expected winning? B) The first box has the highest expected winning. C) The expected winning of the second box is lower than of the third box. D) The expected winning of the third box is higher than of the first box. E) All boxes have the same winnings.
    1. The Pangasinan Provincial Board approved an ordinance of mandatory wearing of faceshield in the province during this public health crisis due to coronavirus disease. What is the mean and standard deviation of the distribution if you know that those who violated the ordinance are being charged with different amounts with the following probabilities? Fine (in pesos) 1,000 2,500 5, Probability 0.25 0.45 0. A) mean of 2875; standard deviation of 1515. B) mean of 2857; standard deviation of 1515. C) mean of 2875; standard deviation of 1515.

premium of P50,000. If she dies within a year, the insurance firm will give her beneficiaries P750,000. According to the 2019 Philippine Statistics Authority Death Statistics Table, the probability that a 35-39 year old woman will not survive in 1 year is 0.0002.

  1. Given X as the net gain of Hermari Life Insurance, which of the following tables of values represents the probability distribution of X? A) C) B) D)
  2. Given that a 35-39 woman will have a 0.0002 chance of not surviving within a year, what is the probability that she will survive within a year? A) 0.998 B) 0.9998 C) 0.99998 D) 0.
  3. What is the expected value of Hermari Life Insurance’s profit on the policy? A) 49,805 B) 49,085 C) 49,850 D) 49,

Additional Activities

Solve the following problems as required.

  1. Based on data from the 2019 Philippine Statistics Authority Death Statistics Table, the probability that a 70-74 year old man will not be alive in 1 year is 0.1028. Assume Sipnayan Life Insurance, an insurance provider, charges P40,000 to cover a 70-74 year old man for the whole year. If he does not live within a year, the insurance firm will pay P380,000 to his survivor as a death benefit. Determine and interpret the mathematical expectation of profit on the policy of the insurance company.
  2. Maria Ophelia Louise intends to invest P500,000 in establishing a company FAROAH clothing line. She estimates that she has a 0. probability of making a P500,000 profit, a 0.25 probability of making a P2,500,000 profit, a 0.35 probability of making no profit. What is the expected value of the profit? Should Maria Ophelia Louise continue on this kind of investment with her clothing line? 23
X 49,980 700,
P(X) 0.9998 0.
X 50,000 700,
P(X)
X 49,980 700,
P(X) 0.998 0.
X 50,000 700,
P(X)

References A. Printed Materials Albert, J.R. F., et al. (2016). Statistics and Probability for Grade 11 Teaching Guide. Pasig City: Department of Education. Belecina, R. R., Baccay, E. S., and Mateo, E.B. (2016). Statistics and Probability, 1st^ Edition. Quezon City: Rex Book Store, Inc. pp. 21- Bluman, A. (2015). Elementary Statistics: A Step by Step Approach. London: McGraw-Hill Education - Europe. pp. 258- Brase, C. H. and Brase, C. P. (2004, 2019). Understanding Basic Statistics 3 rd^ and 8th^ editions. Boston, USA: Cengage Learning. pp. 246- Commission on Higher Education (2016). Teaching Guide for Senior High School: Statistics and Probability Core Subject. Quezon City: CHED. Dela Cruz, E. O., et al. (2016). Soaring 21st^ Century Mathematics: Statistics and Probability (K to 12). Quezon City: Phoenix Publishing House, Inc. pp. 16- Department of Education – Region IV-A CALABARZON (2020). Statistics and Probability Learner’s Material. Cainta, Rizal: DepEd. pp. 80- Department of Education – Schools Division of Bataan (2020). Statistics and Probability Quarter 3 Module 3: The Normal Distribution. Balanga City, Bataan: DepEd. pp. 6 – 21 Marquez, M.P. N. (2005). Statistics in Social Sciences (Slidedeck/handouts). University of the Philippines Population Institute, Diliman Quezon City. Sternstein, M. (2007). Barron’s AP Statistics 4th^ edition. New York: Barron Educational Series. pp. 189 – 219 Triola, M. F. (2010). Elementary Statistics Technology Update 11th^ edition. USA: Pearson Education, Inc. pp. 208 - 218 B. Electronic Sources Department of Information and Communications Technology ( National ICT Household Survey 2019 ). Retrieved on February 1, 2021 from https://dict.gov.ph/ictstatistics/nicths2019/ Metropolitan Community College (2021). Expected Value. Retrieved on February 10, 2021 from https://mcckc.edu/tutoring/blue-river/math.asp Philippine Statistics Authority ( 2019 Death Statistical Tables ). Retrieved on February 4, 2021 from https://psa.gov.ph/civilregistration/statisticsregistered-deaths-philippines- 2019 Rapid Table Page (2020). Variance Calculator. Retrieved on January 28, 2021 from https://www.rapidtables.com/calc/math/variance-calculator.html