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Understanding Statistical Distributions: Discrete and Continuous Random Variables

This comprehensive guide dives into statistical distributions and their applications in probability. Learn about discrete random variables, probability distributions, and key concepts such as the binomial distribution, mean, and standard deviation. Engage with examples and solve exercises to solidify your understanding, including real-life applications of normal distributions. Explore essential topics like the role of sampling, probability density functions, and the significance of the standard normal distribution. Prepare for assessments and enhance your statistical skills through individual and class work.

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Understanding Statistical Distributions: Discrete and Continuous Random Variables

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  1. 29. Statistical distributions. a. SERRA

  2. 29A&B. Discrete random variables and discrete probability distributions. • Reminder : Title > Theory >Examples (if necessary) > Exercises> Correct (✓, ✗,?) 20% hwk> Extra (even + PS + HL) • New: Daily Miniquiz. • Class work: • Continuous vs Discrete: Measuring vs counting. • Discrete random variables and probability distribution associated with it. • EG. Exercise 29B:4 Page 713 • Individual work: • Examples 1,2,3 and 4: Do them & check your answers. • Exercise 29A: 1,3 • Exercise 29B: 3, 7 • Extra: even numbers 29 A and B.

  3. 29C. Expectation. • Class work: • Eg. I am a very good goal keeper . The probability of a penalty kick scoring a goal if I am goalie is only 0.7. How many goals can we EXPECT if 10 LAS students have the chance to try my skills? • Individual work: • Example 5: Do it & check your answers. • Exercise 29C: 1,7, 11, 13 • Extra: Even numbers

  4. 29D. The mean and standard deviation of a discrete random variable. • Class work: • Link from mean (Unit 18) and exercises 29C: 11, 13. • Formulae in blue box in page 719. • Eg. 6 page 721. • Individual work: • Examples 6 and 7: Do them & check your answers. • Exercise 29D: 3,5,7 • Extra: Even numbers

  5. 29 E. The binomial distribution. • Class work: • Sampling with and without replacement: Binomial and the hypergeometric distributions. • Binomial experiment (independent trials with two possible outcomes (yes/no, success/failure, etc.) • Example: • exercise 6 page 724. 2ndVARS (Distr)> Binompdf (20,0.5,10)=0.176 (!) • 2ndVARS (Distr)> Binomcdf(20,0.5,10)=0.588 (!) • Individual work: • Example 8: Do it & check your answers. • Exercise 29 E: 3, 7, 9 • Even numbers.

  6. 29F. Mean and standard deviation of a binomial random variable. • Class work: • Review previous. Formulae page 724. NB Proof not needed. • Example: Exercise 6 in page 727. • Individual work: • Examples 9 and 10: Do them & check your answers. • Exercise 29F: 1 and 5. • Extra: Even numbers

  7. 29G. Normal distributions. • Class work: • Discrete random variables > continuous random variables. • Probability density function. Real life examples (some on page 728). • Geometrical properties and significance of the mean and standard deviation in a bell-shaped curve. (bottom page 729 and 730). • Example: Exercise 6 in page 732. See online videos: • GDC>DISTR>DRAW>ShadeNorm(min,max,mean,sd). • GDC>DISTR>Normalcdf (min,max,mean,sd). • Individual work: • Example 11: Do it & check your answers. • Exercise 29G1: 3 and 7. • Exercise 29G2: 1 • Extra: Even numbers

  8. 29H. The standard normal distribution. • Class work: • The need to have a very special example. Z-distribution. • Models and tables. Page 735. • Example Exercise 29H2.2 Page 738 using table and GDC>DISTR>normalcdf (min,max) • Individual work: • Examples 12, 13 and 14: Do them & check your answers • Exercise 29H1:3 • Exercise 29H2:3 (note that sigma=0.93mm and not 0.93m) • Exercise 29H3:1 • Extra: Even numbers

  9. 29I. Applications of the normal distribution. • Class work: • Example 8 page 741 (see online video). . • Example 2 page 740 Three ways to solve it: • GDC>DISTR>DRAW>ShadeNorm(min,max,mean,sd) • GDC>DISTR>Normalcdf (min,max,mean,sd). • GDC>DISTR>Normalcdf (minZ,maxZ). • Individual work: • Examples 15 and 16: Do them & check your answers. • Exercise 29I: 1,3,5 • Extra: Even numbers

  10. Problem solved step by step (video online) • A student scored 70 for a Science exam and a 66 for Geography. The class scores are normally distributed with a mean and a standard deviation for Science of 60 and 10 and Geography for 50 and 12. • In which subject did the student achieve a higher standard? • What percentage of others achieved lower marks in each subject? • c ) What was the maximum score obtained by the 80% weakest Science students? • What was the minimum score obtained by the top 20% Geography students? • Answer: • a- Science: Z= (70-60)/10 = 1 < Geography Z= (66-50)/12 = 4/3 (aprox. 1.33) • b - Normalcdf (-1000,1)= 0.841 < Normalcdf (-1000,66, 50,12)= 0.909 • c- (X-60)/10 = invNorm (0.8)= 0.842 therefore x=68.4 • d- (X-50)/12 = invNorm (0.8)= 0.842 therefore x=60.1

  11. Review unit 29 • INDIVIDUAL WORKHOMEWORK • Review Set 29A. (Do, correct your answers and write down score (total and percentage “%”) • Mock test: Same • Extra: Create an online quiz using Google forms and share it with the group. Please make sure your answers are correct. A positive in homework and/or in professionalism can be awarded if you do this task! • Next> Test

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