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Continuous Random Variables

Continuous Random Variables. Continuous Random Variables. For discrete random variables, we required that Y was limited to a finite (or countably infinite) set of values. Now, for continuous random variables, we allow Y to take on any value in some interval of real numbers.

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Continuous Random Variables

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  1. Continuous Random Variables

  2. Continuous Random Variables • For discrete random variables, we required that Y was limited to a finite (or countably infinite) set of values. • Now, for continuous random variables, we allow Y to take on any value in some interval of real numbers. • As a result, P(Y = y) = 0 for any given value y.

  3. CDF • For continuous random variables, define the cumulative distribution functionF(y) such that Thus, we have

  4. A Non-Decreasing Function Continuous random variable Continuous distribution function implies continuous random variable.

  5. “Y nearly y” • P(Y = y) = 0 for any y. • Instead, we consider the probability Y takes a value “close to y”, • Compare with density in Calculus.

  6. PDF • For the continuous random variable Y, define the probability density function as for each y for which the derivative exists.

  7. Integrating a PDF • Based on the probability density function,we may write Remember the 2nd Fundamental Theorem of Calc.?

  8. Properties of a PDF • For a density function f(y): • 1).f(y) > 0 for any value of y. • 2).

  9. Problem 4.4 • For what value of k is the following function a density function? • We must satisfy the property

  10. Exponential • For what value of k is the following function a density function? • Again, we must satisfy the property

  11. P(a < Y < b) • To compute the probability of the eventa < Y < b ( or equivalently a< Y <b ),we just integrate the PDF:

  12. Problem 4.4 • For the previous density function • Find the probability • Find the probability

  13. Problem 4.6 • Suppose Y is time to failure and • Determine the density function f (y) • Find the probability • Find the probability

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