Math 3680 Lecture #8 Continuous Random Variables

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.

18. The Uniform Distribution

19. 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.

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

21. The Exponential Distribution

22. 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.

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

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

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

26. 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.