Review Sheet Midterm - Probability and Statistics I - Fall 2008 | MTH 445, Study notes of Probability and Statistics

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11/05/2008 Statistics 445/

STUDY SHEET for MIDTERM (Wednesday, November 12)

The test will cover:

• Ch. 3 (sections 3.4 – 3.9, 3.10)
• Ch. 4 (sections 4.1 – 4.10)
• Ch. 5 (section 5.2)

The best way to prepare to the midterm is to read the book and to do the homework exercises. Please, take time to go over the material.

I. Chapter 3 Discrete Random Variables

1. Some probability distributions: a) Binomial probability distribution; Table 1; b) Geometric probability distribution; c) Negative binomial probability distribution; d) Hypergeometric probability distribution; approximation of hypergeometric distribution by a binomial distribution; e) Poisson probability distribution; Table 3; approximation of a binomial distribution by a Poisson distribution.
2. Moments and moment-generating functions for discrete probability distributions.
3. Tchebysheff’s theorem.

II. Chapter 4 Continuous Random Variables

1. Probability Distribution: a) Distribution function; b) Probability density function; c) Properties.
2. Expected values for continuous random variables.
3. Some continuous probability distributions: a) Uniform probability distribution; b) Normal probability distribution; Standard normal probability distribution; Table 4; c) Gamma probability distribution; d) Exponential probability distribution; the memoryless property of the exponential distribution; e) Chi-square probability distribution; degrees of freedom; Table 6; f) Beta probability distribution.
4. Moments and moment-generating function for continuous probability distributions.
5. Tchebysheff’s theorem.

III. Chapter 5 Multivariate Probability Distributions

1. The joint probability function of two random variables;

2. The joint distribution function of two random variables;
3. The joint probability density function;

Problems:

1. When a white flower is crossed with a red flower, the resulting flower has a 25% chance of being white. Suppose that 20 such flowers are independently produced by cross-breeding. a) Find the mean and variance for the number of white flowers produced; b) Find the probability that exactly 5 white flowers are produced. c) Find the probability that at most 3 white flowers are produced.
2. A personal officer is interviewing job applicants. Each applicant has a probability of 0.2 of being bilingual. a) Find the probability that among 25 applicants at least 5 are bilingual. b) Find the probability that exactly 5 applicants must be interviewed in order to find one who is bilingual.
3. The number of people entering the intensive care unit at a hospital on any single day possesses a Poisson distribution with a mean equal to five persons per day. a) What is the probability that the number of people entering the intensive care unit on a particular day is equal to 2? Is at most 2? b) Is it likely that the number of people entering the intensive care unit on a particular day will exceed 10? Explain.
4. (*) If Y has a Poisson distribution with parameter , show that E(Y)=.
5. Find the distribution, mean, and variance of the random variable that have

as its moment generating function.

6. Let is the moment-generating function for a random

variable Y. Find: a) b) c) Distribution of Y.

7. (*) Derive the moment-generating function for a random variable having a geometric distribution with parameter p.
8. For a certain type of soil the number of wireworms per cubic foot has a mean of 100. Assuming that a Poisson distribution of wireworms, give an interval that will include at least 5/9 of the sample values of wireworm counts obtained from a large number 1- cubic-foot samples.

9. The length of time required by students to complete a 1-hour exam is a random variable ( X ) with a density function given by

otherwise

cx x x f x 0 ,

, 0 1

( )

2

a) Find the constant c so that f ( x )is a probability density function. b) Find the distribution function of X. Graph it. c) Find the probability that a randomly selected student will finish in less than half an hour, P ( X 0. 5 ) d) Find the expected value and the standard deviation of X.

10. A candy maker produces mints that have a label weight of 20.4 grams. Assume that the distribution of these mints is approximately normal with mean 21.37 and standard deviation of 0.16. a) Find the probability that a randomly selected mint has weight anywhere between 21.07 and 21.62 grams. b) If the 2% of the mints with smallest weight is considered to be defective, what is the actual weight for a mint to be considered defective?
11. If Y has an exponential distribution and , what is a) b)
12. If X belongs to 2 distribution with 23 degrees of freedom, find: a) P(14.85 < X < 44.18); b) Constants a and b such that P( a < X < b ) = 0.95 and P( X < a ) = 0.025; c) The mean and the variance of X.
13. (*) If X is , show that is ), a ≠ 0.
14. Find the moment-generating function of the random variable X whose probability density function is given by

otherwise

e x f x

x

0 ,

, 0

( )

and use it to find the mean and standard deviation of X.

15. Find for the uniform random variable with parameters and . Compare with the corresponding probabilistic statements given by Tchebysheff’s theorem and the empirical rule.
16. Do homework for problems from Section 5.2.