1 / 26

Math 3680 Lecture #8 Continuous Random Variables

Learn about continuous random variables, probability density functions, cumulative distribution functions, moments, and distributions like Uniform and Exponential. Explore examples and properties to enhance your understanding.

gamache
Télécharger la présentation

Math 3680 Lecture #8 Continuous Random Variables

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math 3680 Lecture #8 Continuous Random Variables

  2. Suppose we throw a dart at a number line in such a way that it always lands in the interval [1,3]. Let X be the number that the dart hits. Since the possible values of X consist of an entire interval, we call X a continuous random variable. P( cXd ) Discrete Continuous

  3. Definition: A continuous random variable X on an interval [a, b] has a probability density function(pdf) f ( x) that satisfies the following three conditions: 1. f ( x ) ≥ 0 for all values of x. 2. 3. The cumulative distribution function (cdf)of X is FX ( x ) = P( Xx ).

  4. Example:Let X be a random variable with density function f ( x ) = (3 / 14) (x + x2) over [0,2]. A) Verify that f ( x ) is a density function. B) Compute P ( X 1). C) Compute F( x ) for x < 0, for 0 x 2, and for x > 2.

  5. Graph of cumulative distribution function

  6. Properties of cumulative distribution functions • F( x ) = P( Xx ) • 0  F( x )  1 • F is non-decreasing • F is right-continuous: F(x+) = F(x) for each x • For a continuous r.v. X, F is always continuous • For any a < b, P( a < X b ) = F( b ) - F( a )

  7. Theorem:F ’ ( x ) = f ( x ). Proof. By definition, Therefore, by the Fundamental Theorem of Calculus, F ’ ( x ) = f ( x ).

  8. Definition: For a random variable X, the r thpercentile (denoted by xr/100) is the value x so that The idea is that r % of the area under the curve lies to the left of xr/100.

  9. Example: Suppose a random variable X has pdf f ( x ) = over Find the median (50th percentile) of the distribution.

  10. Definition: In statistics, we will often have occasion to compute the critical value that corresponds to a predetermined significance level. The significance level, denoted by a, sets a desired probability, or an area under a tail of the pdf. This kind of calculation is entirely equivalent to finding percentiles.

  11. Example: Suppose a random variable X has pdf f ( x ) = over • Find the right-tail critical value if the significance level is a = 0.01.

  12. Moments

  13. Definition: Var( X ) = E[ (X - m)2 ] s = SD( X ) =  Var( X ) Var( X ) = E[ X 2 ] - m2

  14. Theorem:If a and b are real constants, then E( aX + b ) = a E( X ) + b Var( aX + b ) = a2 Var( X ) SD( aX + b ) = | a | SD( X ) Proof:

  15. Example: Suppose a random variable X has pdf f ( x ) = 6 x - 6 x2 over [0,1]. Find the mean, variance, and standard deviation of X.

  16. Example: A random variable X is said to have the c2(4) distribution if its pdf is Find the mean and standard deviation of X.

  17. The Uniform Distribution

  18. UNIFORM DISTRIBUTION A continuous random variable X is said to have a Uniform(a, b) distribution if its density function is given by for a xb.

  19. Example: Compute the mean and standard deviation of the Uniform(a, b) distribution.

  20. The Exponential Distribution

  21. EXPONENTIAL DISTRIBUTION A continuous random variable X is said to have an Exponential(q ) distribution if its density function is given by for x 0.

  22. Exercise:Confirm that is a probability density function (for x 0).

  23. Exercise: Compute the mean, variance, and standard deviation of the Exponential(q ) distribution.

  24. Memoryless Property of the Exponential Distribution Theorem: Let X be an exponential random variable. Then for all t ≥ 0 and all s ≥ 0, Proof:

  25. Example: In the Luria-Delbrück mutation model, it is assumed that a certain population experiences 0.25 mutations per hour. A mutation just occurred. Compute the probability that • At least four hours pass until the next mutation occurs. • At least four hours, but not more than 8 hours pass until the next mutation occurs. • Repeat (a) and (b), but given that the last mutation occurred three hours ago.

More Related