GENMATH 11 - COMPOSITION OF FUNCTION PPT

lesson discussion

1. COMPOSITE FUNCTIONS By: Ma’am Dolly May S. Palasan

2. PROBABILITY DISTRIBUTION Recognize the composition of functions Perform the composition of functions LEARNING OBJECTIVES At the end of the lesson, the student shall be able to : Appreciate the importance of composite functions in real-life applications.

3. Nonfiction Story : The Discovery of Penicillin by Alexander Fleming In 1928, Alexander Fleming, a scientist working in his laboratory, made a revolutionary discovery by accident. While studying bacteria, he noticed something unusual: a mold had contaminated one of his petri dishes, and where the mold grew, the bacteria were destroyed.

4. Fleming realized that the mold produced a substance that killed bacteria, which he identified as penicillin. In this case, the ‘input’ was the mold, and the ‘output’ was the bacteria being killed. This simple observation led to the development of penicillin, the world’s first antibiotic, which has saved countless lives. Fleming’s discovery demonstrates how an unexpected result from an experiment can lead to a groundbreaking medical advancement.

5. FUNCTIONS A function relates an input to an output. INPUT MACHINE OUTPUT Bread Toaster Toasted Bread Rice grains Rice cooker Cooked Rice Coconut meat Coconut grater Grated Coconut

6. Definition Let f and g be functions The composite function denoted by is defined by     The process of obtaining a composite function is called function composition.

7. Example 1. 𝑝(𝑥)=     Examples: For example 1 and 2 Let, Example 1. Find and simplify Solution.   Example 2. Find and simplify Solution.   𝑓(𝑥)=2𝑥+1 𝑔(𝑥)=√(𝑥+1) Find and simplify              

8. BINOMIAL DISTRIBUTION Example 1: A box of candies has many different colors in it. There is a 15% chance of getting a pink candy. What is the probability that exactly 4 candies in a box are pink out of 10? We have that: n = 10, p=0.15, q=0.85, r=4 When we replace in the formula:                                                                                      Interpretation:  The probability that exactly 4 candies in a box are pink is 0.04.  

9. The binomial distribution is a useful tool in statistical methods of education for a variety of reasons. It can be used to: Calculate the probability of students passing or failing a test. Example. Estimate the mean and standard deviation of students' scores on a test. Compare the performance of different groups of students on a test. Identify students who are at risk of failing a test. A teacher wants to know the probability that her students will pass a test with a 70% passing rate. She can use the binomial distribution to calculate the probability that 70%, 65%, 60%, or any other percentage of her students will pass the test. A school district wants to estimate the mean score of all students in the district on a standardized test. They can use the binomial distribution to estimate the mean score, given the number of students in the district, the percentage of students who passed the test, and the difficulty of the test. A researcher wants to compare the performance of students who received a new teaching method to the performance of students who received a traditional teaching method. They can use the binomial distribution to compare the mean scores of the two groups of students, given the number of students in each group and the percentage of students in each group who passed the test. A school counselor wants to identify students who are at risk of failing a test. They can use the binomial distribution to calculate the probability that a student will fail the test, given the student's past performance on tests, the difficulty of the test, and the student's attendance record.

10. The binomial distribution is a useful tool in statistical methods of education for a variety of reasons. It can be used to: Calculate the probability of students passing or failing a test. Estimate the mean and standard deviation of students' scores on a test. Compare the performance of different groups of students on a test. Identify students who are at risk of failing a test. A teacher wants to know the probability that her students will pass a test with a 70% passing rate. She can use the binomial distribution to calculate the probability that 70%, 65%, 60%, or any other percentage of her students will pass the test. A school district wants to estimate the mean score of all students in the district on a standardized test. They can use the binomial distribution to estimate the mean score, given the number of students in the district, the percentage of students who passed the test, and the difficulty of the test. A researcher wants to compare the performance of students who received a new teaching method to the performance of students who received a traditional teaching method. They can use the binomial distribution to compare the mean scores of the two groups of students, given the number of students in each group and the percentage of students in each group who passed the test. A school counselor wants to identify students who are at risk of failing a test. They can use the binomial distribution to calculate the probability that a student will fail the test, given the student's past performance on tests, the difficulty of the test, and the student's attendance record.

11. BINOMIAL DISTRIBUTION Example 2: A coin is tossed12 times. What is the probability of getting exactly 7 heads? We have that: n = 12, p= 1/2, q=1/2, r=7 When we replace in the formula:                                                                                      Interpretation:  The probability of getting exactly 7 heads is 0.193.  

12. TYPES DISCRETE PROBABILITY DISTRIBUTION BERNOULLI DISTRIBUTION  - is a  discrete probability distribution  for a  Bernoulli trial  — a random experiment that has only two outcomes (usually called a “Success” or a “Failure”). For example, the probability of getting a heads (a “success”) while flipping a coin is 0.5. The probability of “failure” is 1 – P (1 minus the probability of success, which also equals 0.5 for a coin toss). It is a special case of the  binomial distribution  for n = 1. In other words, it is a binomial distribution with a single trial (e.g. a single coin toss).

13. TYPES DISCRETE PROBABILITY DISTRIBUTION BERNOULLI DISTRIBUTION A  Bernoulli trial  is one of the simplest experiments you can conduct. It’s an experiment where you can have one of two possible outcomes. For example, “Yes” and “No” or “Heads” and “Tails.” A few examples: Coin tosses : record how many coins land heads up and how many land tails up. Births : how many boys are born and how many girls are born each day. Rolling Dice : the probability of a roll of two die resulting in a double six.

14. BERNOULLI DISTRIBUTION Example 3. A basketball player can shoot a ball into the basket with a probability of 0.6. What is the probability that he misses the shot? Solution:  We know that success probability P (X = 1) = p = 0.6 Thus, probability of failure is P (X = 0) = 1 - p = 1 - 0.6 = 0.4 Answer:  The probability of failure of the Bernoulli distribution is 0.4

15. TYPES DISCRETE PROBABILITY DISTRIBUTION POISSON DISTRIBUTION -Is a discrete  probability distribution . It gives the probability of an event happening a certain number of times ( k ) within a given interval of time or space. - The Poisson distribution has only one  parameter , λ (lambda), which is the  mean  number of events. -it is use to predict or explain the number of events occurring within a given interval of time or space. “Events” could be anything from disease cases to customer purchases to meteor strikes. The interval can be any specific amount of time or space, such as 10 days or 5 square inches .

16. POISSON DISTRIBUTION You can use a Poisson distribution if: Individual events happen at random and independently. That is, the probability of one event doesn’t affect the probability of another event. You know the  mean  number of events occurring within a given interval of time or space. This number is called λ (lambda), and it is assumed to be constant. For example, a Poisson distribution could be used to explain or predict: Text messages per hour Male grizzly bears per hectare Machine malfunctions per year Website visitors per month Influenza cases per year

17. POISSON DISTRIBUTION Poisson distribution formula The probability mass function of the Poisson distribution is: Where:   is a random variable following a Poisson distribution   is the number of times an event occurs is the probability that an event will occur  k  times   is Euler’s constant (approximately 2.718)   is the average number of times an event occurs ! is the factorial function  

18. POISSON DISTRIBUTION Example 4. An average of 0.61 soldiers died by horse kicks per year in each Prussian army corps. You want to calculate the probability that exactly two soldiers died in the VII Army Corps in 1898, assuming that the number of horse kick deaths per year follows a Poisson distribution.  = 2 deaths by horse kick  = 0.61 deaths by horse kick per year  = 2.718 The probability that exactly two soldiers died in the VII Army Corps in 1898 is 0.101.

19. CONTINOUS PROBABILITY DISTRIBUTION -describes the probabilities of a continuous random variable's possible values. A continuous random variable has an infinite and uncountable set of possible values (known as the range). Properties of a continuous probability distribution include: The outcomes are measured, not counted. The entire area under the curve and above the x-axis is equal to one. Probability is found for intervals of x values rather than for individual  x   values. P ( c < x < d )   is the probability that the random variable X is in the interval between the values c and d.  P ( c < x < d ) is the area under the curve, above the x-axis, to the right of  c   and the left of  d . P ( x = c )=0   The probability that  x   takes on any single individual value is zero. The area below the curve, above the x-axis, and between  x = c   and  x = c   has no width, and therefore no area ( area =0 area =0 ). Since the probability is equal to the area, the probability is also zero. P ( c < x < d )   is the same as  P ( c ≤ x ≤ d )   because probability is equal to area.

20. CONTINOUS PROBABILITY DISTRIBUTION Example 1. Consider the function f(x)120 is a horizontal line. However, since 0≤x≤20,  f ( x ) is restricted to the portion between x=0 and x=20, inclusive. f(x)=120 for 0≤x≤20 The graph of f(x)=120 is a horizontal line segment when 0≤x≤20. The area between f(x)120. AREA=20(1/20)=1

21. TYPES CONTINOUS PROBABILITY DISTRIBUTION NORMAL DISTRIBUTION -is also known as the Gaussian distribution, is a  probability distribution  that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. -In graphical form, the normal distribution appears as a " bell curve ". mean  =  median  =  mode symmetry about the center 50% of values less than the mean and 50% greater than the mean Mean Median Mode 50% 50%

22. CHARACTERISTICS OF NORMAL DISTRIBUTION Empirical Rule : In a normal distribution, 68% of the observations are confined within -/+ one standard deviation, 95% of the values fall within -/+ two standard deviations, and almost 99.7% of values are confined to -/+ three standard deviations. Bell-shaped Curve : Most of the values lie at the center, and fewer values lie at the tail extremities. This results in a bell-shaped curve. Mean and Standard Deviation : This data representation is shaped by mean and standard deviation. Equal Central Tendencies : The mean, median, and mode of this data are equal. Symmetric : The normal distribution curve is centrally symmetric. Therefore, half of the values are to the left of the center, and the remaining values appear on the right. 

23. CHARACTERISTICS OF NORMAL DISTRIBUTION Skewness and Kurtosis : Skewness is the symmetry. The skewness for a normal distribution is zero. If the normal distribution is uneven with a skewness greater than zero or positive skewness , then its right tail will be more prolonged than the left. Similarly, for positive skewness the left tail will be longer than the right tail. Negative skewness means skewness is less than zero. Kurtosis  is a measure of peakiness . If the kurtosis is 3, the probability data is neither too peaked nor too thin at tails. If the kurtosis is more than three, then the data curve is heightened with fatter tails. Alternatively, if the kurtosis is less than three, then the represented data has thin tails with the peak point lower than the normal distribution. Kurtosis studies the tail of the represented data. For a normal distribution, the kurtosis is 3. Total Area = 1 : The total value of the standard deviation, i.e., the complete area of the curve under this probability function, is one. Also, the entire mean is zero.

24. FORMULA OF NORMAL DISTRIBUTION The Probability Density Function (PDF) of a random variable (X) is given by: Where; -∞ < x < ∞; -∞ < µ < ∞; σ > 0 F(x) = Normal probability Function x = Random variable µ = Mean of distribution σ = Standard deviation of the distribution π = 3.14159 e = 2.71828  

25. Example 4. Calculate the probability density function of normal distribution using the following data. x = 3, μ = 4 and σ = 2. Solution: Given, variable, x = 3 Mean = 4 and Standard deviation = 2 By the formula of the probability density of normal distribution, we can write;          

26. NORMAL DISTRIBUTION 68%  of values are within 1 standard deviation  of the mean +1 -1 +2 -2 +3 -3 95%  of values are within 2 standard deviations  of the mean 99.7%  of values are within 3 standard deviations  of the mean The  Standard Deviation  is a measure of how spread out numbers are It is good to know the standard deviation, because we can say that any value is: likely  to be within 1 standard deviation (68 out of 100 should be) very likely  to be within 2 standard deviations (95 out of 100 should be) almost certainly  within 3 standard deviations (997 out of 1000 should be)

27. NORMAL DISTRIBUTION Example 2: 95% of students at school are between  1.1m and 1.7m  tall. 1 standard deviation = (1.7m-1.1m) / 4   = 0.6m / 4   =  0.15m Assuming this data is  normally distributed  can you calculate the mean and standard deviation? The mean is halfway between 1.1m and 1.7m: Mean = (1.1m + 1.7m) / 2 =  1.4m 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so: And this is the result:

28. The number of  standard deviations from the mean  is also called the "Standard Score", "sigma" or "z-score". is the z-score is the value to be standardized   26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34 The  z-score formula  is:   Example 3: Travel Time A survey of daily travel time had these results (in minutes): = 38. 8 minutes = 11. 4   is the mean is the standard deviation      

29. Example 4. Let us suppose that a company has 10000 employees and multiple salary structures according to specific job roles. The salaries are generally distributed with the  population mean  of µ = $60,000, and the population standard deviation σ = $15000. What will be the probability of a randomly selected employee earning less than $45000 per annum? Given, Mean (µ) = $60,000 Standard deviation (σ) = $15000 Random Variable (x) = $45000       Thus, it indicated that when we randomly select an employee, the probability of making less than $45000 a year is 15.87%.

30. SAMPLING DISTRIBUTION

31. SAMPLING DISTRIBUTION