More Related Content
Similar to wk-2.pptx (20)
Recently uploaded (20)
wk-2.pptx
- 1. MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE
- 2. A. 4 5 5 7 8 9 B. 1 2 2 8 11 14 CONSIDER THE FOLLOWING DATA:
- 3. A. 4 5 5 7 8 9 B. 1 2 2 8 11 14 FIND THE MEAN.
- 4. A. 4 5 5 7 8 9 B. 1 2 2 8 11 14 FIND THE MEAN. =6.33 =6.33
- 5. A. 4 5 5 7 8 9 B. 1 2 2 8 11 14 Describe the mean. Compare the data presented. =6.33 =6.33
- 6. MEAN Average The sum of a collection of numbers divided by the count of numbers in the collection.
- 7. VARIANCE Dispersion of data in the data set.
- 8. STANDARD DEVIATION Average amount of variability in the dataset. It tells us on how far each value lies from the mean.
- 9. Therefore, the variance of the probability distribution is 2.69.The standard deviation is = 2.69 = . Number of Cars Sold Probability () () ( ) ( ) () 1 1 X = 1 3.1 = -2.1 (. ) = . . = . 2 2 X = 2 3.1 = -1.1 (. ) = . . = . 3 3 X = 3 3.1 = -0.1 (. ) = . . = . 4 4 X = 4 3.1 = 0.9 (. ) = . . = . 5 5 X = 5 3.1 = 1.9 (. ) = . . = . = X P(X) = = 3.1 = ( ) = .
- 10. Assessment: Choose the letter of the correct answer. 1. Which of the following statements is TRUE about the interpretation of the values of variance and standard deviation? a. A small value of variance or standard deviation indicates that the distribution of the discrete random variable is closer about the mean. b. A large value of variance or standard deviation indicates that the distribution of the discrete random variable is closer about the mean. c. A small value of variance or standard deviation indicates that the distribution of the discrete random variable takes some distance from the mean. d. All of the above.
- 11. 2. In 50 items test, Miss Santos, a mathematics teacher claimed that most of the students scores lie closer to 35. In this situation, score of 35 is considered as, A. Variance C. Expected Value or Mean B. Standard Deviation D. Median
- 12. 3. Which of the following statement describes variance of a discrete random variable? A. It is a weighted average of the possible values that the random variable can take. B. It is the product of mean and the square of the probability distribution of a discrete random variables. C. It is obtained by getting the summation of the product of the square of the difference between the value of X and the expected value times its corresponding probability D. All of the above
- 13. 4. If P(X) = , what are the possible values of X for it to be a probability distribution? A. 0, 2, 3 C. 2, 3, 4 B. 1, 2, 3 D. 1, 1, 2
- 14. 5. Which of the following statements is NOT TRUE about variance? A. cannot be negative B. greater than 0 C. less than 0 D. a measure of spread for a distribution of a random variable
- 15. 6. It is a weighted average of the possible values that the random variable can take. A. Mean B. Variance C. Standard Deviation D. Probability Distribution
- 16. 7. The appropriate formula in finding the mean of discrete random variable is A. E x = 亮x = x p (x) B. E x = 亮x = x + p (x) C. E x = 亮x = x p (x) D. E x = 亮x = x p (x)2
- 17. 8. What formula is used to find the variance of discrete random variable? A. 2 = ( + )2 (); for all possible values of x B. 2 = ( )2 (); for all possible values of X C. 2 = (); for all possible values of x D. 2 = (() + )2 ; for all possible values of x
- 18. EXERCISES Find the mean, variance and standard deviation of the discrete random variable X whose probability distribution is (X) PROBABILITY P(X) 1 0.21 2 0.34 3 0.24 4 0.21
- 19. NORMAL PROBABILITY DISTRIBUTION It is a probability distribution of a continuous random variables. It shows graphical representations of random variables obtained through measurement like the height and weight of the students.
- 21. NORMAL PROBABILITY DISTRIBUTION This graphical representation is popularly known as a normal curve. Normal curve is described clearly by the following properties. Properties of Normal Curve 1. The distribution curve is bell-shaped. 2. The curve is symmetrical about its center.
- 22. NORMAL PROBABILITY DISTRIBUTION This graphical representation is popularly known as a normal curve. Normal curve is described clearly by the following properties. Properties of Normal Curve 3. The mean, the median and the mode coincide at the center. 4. The width of the curve is determined by the standard deviation of the distribution
- 23. NORMAL PROBABILITY DISTRIBUTION This graphical representation is popularly known as a normal curve. Normal curve is described clearly by the following properties. Properties of Normal Curve 5. The tails of the curve flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve is asymptotic to the base line.
- 24. NORMAL PROBABILITY DISTRIBUTION This graphical representation is popularly known as a normal curve. Normal curve is described clearly by the following properties. Properties of Normal Curve 6. The area under the curve is 1. Thus it represents the probability or proportion or the percentage associated with specific sets of measurement values.
- 25. NORMAL PROBABILITY DISTRIBUTION When the normal probability distribution has a mean 袖 = 0 and standard deviation 董 = 1, it is called as standard normal distribution.
- 26. EMPIRICAL RULE The diagram shows the representation of 68% - 95% - 99.7% rule. The 68% -95% - 99.7% rule is better known as empirical rule.
- 27. EMPIRICAL RULE This rule states that the data in the distribution lies within the 1, 2, and 3 of the standard deviation of the mean. Specifically, the above diagram tells the estimation of the following percentage:
- 28. EMPIRICAL RULE 68% of data lies within the 1 standard deviation of the mean. 95% of data lies within the 2 standard deviation of the mean. 99.7% of data lies within the 3 standard deviation of the mean.
- 29. NORMAL PROBABILITY DISTRIBUTION Example: The score of the Senior High School students in their Statistics and Probability quarterly examination are normally distributed with a mean of 35 and standard deviation of 5.
- 30. NORMAL PROBABILITY DISTRIBUTION Example: The score of the Senior High School students in their Statistics and Probability quarterly examination are normally distributed with a mean of 35 and standard deviation of 5. Answer the following questions: What percent will fall within the score 30 to 40? What scores fall within 95% of the distribution?
- 31. NORMAL PROBABILITY DISTRIBUTION Solution: Draw a standard normal curve and plot the mean at the center. Then, add five times the given standard deviation to the right of the mean and subtract 5 times to the left. The illustration is provided below:
- 32. NORMAL PROBABILITY DISTRIBUTION Answer: The scores 30 to 40 falls within the first standard deviation of the mean. Therefore, it is approximately 68% of the distribution Since 95% lies within the 2 standard deviation of the mean, then the corresponding scores of this distribution are from 25 up to 45
- 33. NORMAL PROBABILITY DISTRIBUTION Example 2. What is the frequency and relative frequency of babies weights that are within: 6.11 mean/ 1.63 standard deviation a. One standard deviation from the mean b. Two standard deviation from the mean 2.24 4.21 5.16 5.63 6.18 6.4 6.8 7.34 8.47 2.93 4.38 5.26 5.84 6.19 6.56 6.83 7.35 8.6 3.45 4.69 5.32 5.87 6.24 6.61 7.19 7.35 9.01 3.99 4.94 5.37 6.11 6.38 6.76 7.29 7.68 9.47
- 34. NORMAL PROBABILITY DISTRIBUTION Example 2. What is the frequency and relative frequency of babies weights that are within: a. One standard deviation from the mean b. Two standard deviation from the mean 2.24 4.21 5.16 5.63 6.18 6.4 6.8 7.34 8.47 2.93 4.38 5.26 5.84 6.19 6.56 6.83 7.35 8.6 3.45 4.69 5.32 5.87 6.24 6.61 7.19 7.35 9.01 3.99 4.94 5.37 6.11 6.38 6.76 7.29 7.68 9.47 26 out of 36 or 72% 34 out 36 or 95%
- 35. A. Directions: True or False. In the answer sheet, write the word TRUE if the statement is correct and FALSE, if the statement is incorrect. _______1. The total area of the normal curve is 1. _______2. The normal probability distribution has a mean 袖 = 1 and standard deviation 董 = 0. _______3. The normal curve is like a bell-shaped. _______4. The curve of a normal distribution extends indefinitely at the tails but does not touch the horizontal axis. _______5. About its mean 0, the normal curve is not symmetrical to the center.
- 36. B. Read the following problems carefully. Use empirical rule to answer each question. 1. IQ scores of the ALS students in the Division of Bohol are normally distributed with a mean of 110 and a standard deviation of 10. What percent of the distribution falls within the IQ scores of 100 to 130? 2. A normal distribution of data with the mean of 78 and standard deviation of 9. What percentage of the data would measure 87?