1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Review of Exam 2 Sections 4.6 – 5.6 Jiaping Wang Department of Mathematical Science 04/01/2013, Monday

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Outline Negative Binomial, Poisson, Hypergeometric Distributions and Moment Generating Function Continuous Random Variables and Probability Distribution Uniform, Exponential, Gamma, Normal Distributions

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 1. Part 1. Negative Binomial, Poisson, Hypergeometric Distributions and MGF

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Negative Binomial Distribution If r=1, then the negative binomial distribution becomes the geometric distribution. What if we were interested in the number of failures prior to the second success, or the third success or (in general) the r-th success? Let X denote the number of failures prior to the r-th success, p denotes the common probability. What if we were interested in the number of failures prior to the second success, or the third success or (in general) the r-th success? Let X denote the number of failures prior to the r-th success, p denotes the common probability.

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Poisson Distribution Recall that λ denotes the mean number of occurrences in one time period, if there are t non-overlapped time periods, then the mean would be λt. Poisson distribution is often referred to as the distribution of rare events. E(X)= V(X) = λfor Poisson random variable.

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Hypergeometric Distribution Now we consider a general case: Suppose a lot consists of N items, of which k are of one type (called successes) and N-k are of another type (called failures). Now n items are sampled randomly and sequentially without replacement. Let X denote the number of successes among the n sampled items. So What is P(X=x) for some integer x?

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Moment Generating Function It often is easier to evaluate M(t) and its derivatives than to find the moments of the random variable directly.

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 2. Part 2. Continuous Random Variables and Probability Distribution

9. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Density Function A random variable X is said to be continuous if there is a function f(x), called probability density function, such that Notice that P(X=a)=P(a ≤ X ≤ a)=0. A random variable X is said to be continuous if there is a function f(x), called probability density function, such that Notice that P(X=a)=P(a ≤ X ≤ a)=0.

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Distribution Function The distribution function for a random variable X is defined as F(b)=P(X ≤ b). If X is continuous with probability density function f(x), then Notice that F’(x)=f(x). The distribution function for a random variable X is defined as F(b)=P(X ≤ b). If X is continuous with probability density function f(x), then Notice that F’(x)=f(x). For example, we are given Thus,

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Expected Values

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Variance

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 3. Part 3. Uniform, Exponential, Gamma, Normal Distributions

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Uniform Distribution – Density Function

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Uniform Distribution – CDF

16. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Uniform Distribution -- Mean and Variance

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Probability Density Function θ = 2 θ = 1/2

18. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Cumulative Distribution Function θ = 2 θ = 1/2

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Mean and Variance

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Probability Density Function (PDF)

21. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

22. Probability Density Function

23. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Standard Normal Distribution Let Z=(X-μ)/σ, then Z has a standard normal distribution It has mean zero and variance 1, that is, E(Z)=0, V(Z)=1. Let Z=(X-μ)/σ, then Z has a standard normal distribution It has mean zero and variance 1, that is, E(Z)=0, V(Z)=1.

24. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Mean and Variance

25. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL For example, P(-0.53<Z<1.0)=P(0<Z<1.0) +P(0<Z<0.53)=0.3159+0.2019 =0.5178 P(0.53<Z<1.2)=P(0<Z<1.2)- P(0<Z<0.53)=0.3849-0.2019 =0.1830 P(Z>1.2)=1-P(Z<1.22)=1- 0.3888=0.6112