Probability Distribution

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1 Lecture 4 Probability Distribution Continuous Case Definition: A random variable that can take on any value in an interval is called continuous. Definition: Let Y be any r.v. The distribution function of Y, F(y), is such that. Example: Y ~ Bin(2, ½). Find F(y).

called continuous. Definition: Let Y be any r.v.

2 Properties of Distribution Function: F(y) is a non-decreasing function of y Definition: A r.v. Y with distribution function F(y) is said to be continuous if F(y) is continuous, for. Note: If Y is a continuous r.v. then for any. Definition: Let F(y) be the distribution function for a continuous r.v. Y. Then (wherever the derivative exists), is called the probability density function for Y.

Y with distribution function F(y) is said to be continuous if F(y) is continuous, for.

3 Properties of Density Function: Example: Given {, find f(y).

4 Example: Let Y be a continuous r.v. with {. Find F(y). Theorem: If Y is a continuous r.v. with f(y) and a < b, then Proof:.

5 Example: (#4.16) { is the density function for Y. (a) Find c and F(y); (b) Graph f(y) and F(y); (c) Find.

6 Expected Value Definition: The expected value of a continuous r.v. Y is (provided ) Theorem: Let g(y) be a function of Y, then [ ] (provided ) Properties of Expected Value: 1. E(c) = c, c is a constant. 2. E[cg(Y)]=cE[g(Y)], g(y) is a function of Y. 3. [ ] [ ] [ ], are functions of Y. Proof:

7 Example: Given {, find E(Y) and Var(Y).

8 Uniform Distribution Definition: If, a r.v. Y is said to have a continuous uniform probability distribution on the interval if and only if the density function of Y is {. Theorem: If Y ~ Unif, then and Proof:.

10 Example: (#4.44) The change in depth of a river from one day to the next, measured at a specific location is a r.v. Y with (a) Find k; (b) What s the distribution function of Y? { Example: (#4.50) Beginning at 12:00 am, a computer center is up for 1 hour and then down for 2 hours on a regular cycle. A person who is unaware of the schedule dials the center at a random time between 12:00 am and 5:00 am. Find P(center is up when call comes in).

{ 50) Beginning at 12:00 am, a computer center is up for 1 hour and then down for 2 hours on a regular cycle.

11 Normal Distribution Definition: A r.v. Y is said to have a normal probability distribution if and only if, for and,. Theorem: If, then and. Proof: later. Example: Z~N(0, 1) (a) Find ; (b) Find.

12 Example: (#4.68) The grade point averages (GPAs) of a large population of students are appr. Normally distributed with and. What fraction of the students will possess a GPA greater than 30? Example: (#4.73) The width of bolts of fabric is normally distributed with and. What is the probability that a randomly chosen bolt has a width of between 947 and 958?

$What fraction of the students will possess a GPA greater than 30?$

13 Gamma Distribution Definition: A r.v. Y is said to have a gamma distribution with parameters and if and only if { Where is the gamma function. Properties of Gamma Function:

14 Theorem: If, then and. Proof:

15 Definition: Let. A r.v. Y is said to have a Chi-square ( ) distribution with degrees of freedom if and only if. Theorem: If, then and. Proof: Definition: A r.v. Y is said to have an exponential distribution with parameter if and only if. Theorem: If, then and. Proof:

16 Example: (#4.88) The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distribution with mean 2.4. Find the probability that an earthquake striking this region will (a) exceed 3.0; (b) fall between 2.0 and 3.0. Example: (#4.96) Given { (a) Find k; (b) Does Y have a -distribution? If so, what is? (c) Find E(Y) and Var(Y).

having an exponential distribution with mean 2.4.

17 Moments and Moment-Generating Functions Definition: If Y is a continuous r.v., then the the origin is, k=1,2, moment about The moment about the mean, or the central moment, is, k=1,2, Example: Find for.

18 Definition: If Y is a continuous r.v., then the moment-generating function (mgf) of Y is The mgf exists if there is a constant b > 0 such that m(t) is finite for..

19 Example: Let. Find mgf for Y.

20 Theorem: Let Y be a r.v. with density function f(y) and g(y) be a function of Y. Then the mgf of g(y) is [ ]. Example: Let,. Find mgf for g(y).

21 Example: (#4.137) Show that if Y is a r.v. with mgf m(t) and U is given by U = ay + b, then the mgf of U is.. Find E(U) and Var(U), given that and. Example: (#4.138) Let. Find.

PSTAT 120B Probability and Statistics

PSTAT 120B Probability and Statistics - Week University of California, Santa Barbara April 10, 013 Discussion section for 10B Information about TA: Fang-I CHU Office: South Hall 5431 T Office hour: TBA email: chu@pstat.ucsb.edu Slides will