Probability Distribution, Mean, and Variance PDF

STATISTICS Probabil AND PROBABILITY First Semester ityDistri bution September 16, 2024 Ms. Arlene A. Aboga, MEd Mean and Variance of Random Variables  MEAN OF DISCRETE RANDOM...

STATISTICS Probabil AND PROBABILITY First Semester ityDistri bution September 16, 2024 Ms. Arlene A. Aboga, MEd Mean and Variance of Random Variables  MEAN OF DISCRETE RANDOM VARIABLES Definition VARIANCE If P(X=x) is the probability of every value of x, 𝜇=∑ [ 𝑥∙ 𝑃(𝑥) ] Mean Formula 𝜎 =∑ ( 𝑥−𝜇 ) ∙ 𝑃(𝑥) Variance 2 2 Standard Deviation  Mean or Expected Value of Discrete Probability Distribution Determine the Mean or Expected Value of each Random Variable. Determine the Mean or Expected Value of each Random Variable. Example 1 If two coins are tossed, the possible outcomes are HH, HT, TH, or TT. If X is the random variable for the number of head, Find the mean of the Example 1 If two coins are tossed, the possible outcomes are HH, HT, TH, or TT. If X is the random variable for the number of head, Find the mean of the probability distribution. Example 2 What is the Solution mean outcome if a fair die is rolled? Let Y be the random variable defined by the outcomes Example 3 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Example 2 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Solution Example 2 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Solution  Example 2 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Solution  Example 2 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Solution  Example 2 Your father said, if one grading period your grade in Math is 90 and above, he will add 50qr to your daily allowance, 20qr if your grade is 80 – 89, but decrease it by 10qr if your grade is 79 and below. If the probability to get 90 and above is 12%, while you have 45% chance to get 80 – 89, what is your expected additional allowance per quarter? Solutio n Example 2 Janus is paid 20Qr whenever the result of tossing two coins Solution are both heads, but pays 10QR whenever the results are not both heads. What is his expected gain per toss?  STATISTICS Binomi AND PROBABILITY First Semester al Distribu tion Ms. Arlene A. Aboga, MEd  BINOMIAL DISTRIBUTION A binomial distribution is a Definitio probability with only two possible n outcomes: success Let p andof success, q = = probability probability of failure, and n = failure number of trials performed during an experiment of getting x successes. The probability of getting x successes out of n trials is given by The probability mass function of a binomial distribution is given by  BINOMIAL DISTRIBUTION Definitio n  Formula EXAMPLE: Consider tossing a SOLUTIO 6-sided die three N: Let p = probability that 4 will times. Determine turn up, p q = probability = 1/6 that 4 fails to the probability n = 3 turn up, q = 5/6 that 4 will turn up: tosses 1) 0 times 2) once 3) twice 4) thrice  Formula EXAMPLE: Consider tossing a SOLUTIO 6-sided die three N: Let p = probability that 4 will times. Determine turn up, p q = probability = 1/6 that 4 fails to the probability n = 3 turn up, q = 5/6 that 4 will turn up: tosses 1) 0 times 2) once 3) twice 4) thrice  Formula EXAMPLE: Consider tossing a SOLUTIO 6-sided die three N: Let p = probability that 4 will times. Determine turn up, p q = probability = 1/6 that 4 fails to the probability n = 3 turn up, q = 5/6 that 4 will turn up: tosses 1) 0 times 2) once 3) twice 4) thrice  Formula EXAMPLE: Consider tossing a SOLUTIO 6-sided die three N: Let p = probability that 4 will times. Determine turn up, p q = probability = 1/6 that 4 fails to the probability n = 3 turn up, q = 5/6 that 4 will turn up: tosses 1) 0 times 2) once 3) twice 4) thrice  Example 2 Five friends attended the volleyball varsity tryouts. What is the probability that only three of them will be selected if P = 60% and Q = 40% for each of them?  Example 2 Five friends attended the volleyball varsity tryouts. What is the probability that only three of them will be selected if P = 60% and Q = 40% for each of them?  On your notebook! The probability that a person can achieve a target is 3/4. The count of tries is 5. What is the probability that he will attain the target at least thrice?  Activity 2 Binomial Distribution 2) A bag contains 3) Charlotte 10 marbles, 6 of 1) which are blue and Clara are planning a 7 Binomial and 4 are red. An day trip to experiment probabilit consists of picking Dubai, UAE. y a marble (at The probability random) from the that it rains on distributi bag, making a any one of on with n note of its color those days is = 25, and and putting it 0.3 What is the back in the bag. probability p = 0.8. This experiment is that it rains  Variance of Discrete Probability Distribution The variance is a measure of spread of dispersion. It measures the variation of the values of a random variable used for the variance is 𝜎^2 from the mean. The symbol and its square root  is called the standard deviation.  Students were asked to estimate the length (in inches) of a table. The errors in the estimated values were recorded and tabulated as follows: 1) Find the mean error of this probability distribution. 2) Determine the variance and the standard deviation.  Students were asked to estimate the length (in inches) of a table. The errors in the estimated values were recorded and tabulated as follows: 1) Find the mean error of this probability distribution. 2) Determine the variance and the standard deviation.  Students were asked to estimate the length (in inches) of a table. The errors in the estimated values were recorded and tabulated as follows: 1) Find the mean error of this probability distribution. 2) Determine the variance and the standard deviation.  Students were asked to estimate the length (in inches) of a table. The errors in the estimated values were recorded and tabulated as follows: 1) Find the mean error of this probability distribution. 2) Determine the variance and the standard deviation. Additional Word problems Example 1 Example 1 Example 2  Example 2 a. What is the probability that the company will NOT pay any amount to its client within a year? Example 2 a. What is the probability that the company will NOT pay any amount to its client within a year? 1 - 0.003 = 0.997  Example 2 a. What is the probability that the company will NOT pay any amount to its client within a year? 1 - 0.003 = 0.997 b. What is the expected value of the insurance to its buyer?  Example 2 a. What is the probability that the company will NOT pay any amount to its client within a year? 1 - 0.003 = 0.997 b. What is the expected value of the insurance to its buyer? x 575 000 -25 000 P(x) 0.003 0.997 Example 3 Example 3 1st prize 2nd prize 3rd prize net loss x 9, 950 4, 950 2, 450 -50 P(x) 1/1000 1/1000 1/1000 997/1000 Example 4 Activity 3 - Word Problems The number of cars sold per day at a local 1 dealership, along with its corresponding car probabilities, is shown in the table below. Compute the variance and the standard deviation of the probability distribution. Number of cars sold (x) Probability P(x) 0 1/10 1 2/10 2 3/10 3 2/10 4 2/10  2 A local club plans to invest P10000.00 to host a baseball game. They expect to sell tickets worth P15000.00. But if it rains on the day of the game, they won't sell any tickets and the club will lose all the money instead. If the weather forecast for the day of the game is 20% possibility of rain, will this be a good investment? Why or why not?  3 A company makes electronic gadgets. One out of every 50 gadgets is faulty, but the company doesn't know which ones are faulty until a buyer complains. Suppose the company makes a 3QR profit on the sale of any working gadget, but suff ers a loss of 80QR for every faulty gadget because they need to repair the unit. Determine whether the company can expect a profit in the long run. BOOK ACTIVITY Answer p. 27 Learning E nhancement Worksheet A, B, and C Answer p. 35 Learning E nhancement Worksheet A, B, and C  Study for QUIZ 2! Probability Distribution , Mean and Variance Population Mean When finding the mean without the probability or the P(x), we use The population mean is an average of a group characteristic. The group could be a person, item, or thing, like “all the people living in Qatar” or “all dog owners in Manila”. A characteristic is just an item of interest. The Population Mean concept comes under parent topic statistics. In brief statistics deals with the topics of collecting data, analyzing, interpreting, and presenting the organized data. Population Mean Formula is the average of all the values in the given data/population. Formula The formula to find the population mean is: μ = ΣX/N where: Σ means “the sum of.” X = all the individual items in the group. N = the number of items in the group. Question 1: What is the population mean for the given weights of persons- {50, 60, 70, 100, 80, 55, 60, 65} Question 2: Find the population mean for the given population 10, 20, 30, 40, 50, 55, 45, 35, 25, 15. Question 3: Find the population mean for the given population 20, 25, 20, 25, 22. Formula: The formula to find the population variance is: where: Σ means “the sum of.” X = all the individual items in the group. N = the number of items in the group.  Exampl e 1: 1) Find the mean and the population variance of the age of children in a family of five children aged 16, 11, 9, 8, and 1.  Exampl e 2: 1) Find the population variance of the age of children in a family of five children aged 16, 11, 9, 8, and 1.  Exampl e 3: 2) Find the population variance of the following data set: 11, 12, 55, 4, 17, 13, 19  Exampl e 4: 3) Find the population variance of the following data set: 12, 17, 18, 21