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Understanding Exponential, Logarithmic, and Logistic Growth Models in Statistics

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Explore the fundamental concepts of exponential growth, decay, and logistic growth models. Discover how the Gaussian model, also known as the bell curve, applies to real-valued random variables and their clustering around a mean. The exponential equations illustrate growth and decay dynamics, while the logistic model captures population growth in an S-shaped curve, demonstrating initial rapid growth followed by saturation. Learn about the equations and applications of these models in real-world scenarios, including population dynamics and statistical analysis.

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Understanding Exponential, Logarithmic, and Logistic Growth Models in Statistics

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  1. 3.5 Exponential and Logarithmic Models Gaussian Model Logistic Growth model Exponential Growth and Decay

  2. Gaussian Model or the Bell curve The normal (or Gaussian) distribution is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function is "bell"-shaped, and is known as the Gaussian function or bell curve:

  3. Gaussian Model or the Bell curve If I was curving your grades, 68.2% of the students would have a C, 13.6% a B or D and 2.1% a A or F. 0.1% would have an A+

  4. Gaussian Model or the Bell curve Its equations would be y = ae-[(x –b)^2]/c , where a ,b and c are real numbers.

  5. y = ae-[(x –b)^2]/c Let a = 4; b = 2 and c = 3. The graph will never touch the x axis.

  6. Exponential Growth/ Decay models Growth equation y = aebx b> 0 Decay equation y = ae-bx b>0 Both these models we have seen before in Algebra 2 and in Pre- Cal

  7. Growth equation y = aebx Let a = 5 and b = 2

  8. Decay equation y = ae-bx Let a = 2 and b = 2

  9. Will a small lake have exponential growth of game fish forever? No, What are the factors that keep the lake from the lake filling up with fish?

  10. Logistic growth model • A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve (abbreviated S-curve) of growth of some population P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops. Pierre Francois Verhuist http://en.wikipedia.org/wiki/Logistic_function

  11. Logistic Growth Model a, b and r are positive numbers. a is the maximum limit of the function.

  12. Logistic Growth Model Let a = 10, b = 4 and r = 2

  13. Homework Page 243- 248 # 18, 25, 28, 29, 35, 40 , 47, 50, 63, 70, 74, 93

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