PROBABILITY NOTES - PR2
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Random variable ? : A random variable is a function on a sample space : . This function assigns a real number ? to each sample point : . Often a random variable is simply equal to the
PROBABILITY NOTES - PR4
JOINT, MARGINAL, AND CONDITIONAL DISTRIBUTIONS
Joint distribution of random variables ? and @ : A joint distribution of two random variables has a probability function or probability density function % & that is a function of two v
PROBABILITY NOTES - PR3
FREQUENTLY USED DISCRETE DISTRIBUTIONS
Uniform distribution on 5 points (where 5 is an integer): The probability function is % ~ 5 for % ~ 5 , and % ~ otherwise.
5 ! ! 5 ! c b c ,? ~ 5 , = ? ~ 5 , 4? ! ~ ~ 5 ! c for any real !. ~ 5
PROBABILITY NOTES - 1 SAMPLE SPACES AND EVENTS
Sample point and sample space: A sample point is the simple outcome of a random experiment. The sample space is the collection of all sample points related to a specified experiment. Mutually exclusive outcom
PROBLEM SET 8
1. Let ? and @ be continuous random variables having a bivariate normal distribution with means ? and @ , common variance , and correlation coefficient ?@ . Let -? and -@ be the cumulative distribution functions of ? and @ respectively. Dete
ACT245H1 S - PROBLEM SET 6
1. Let ? be a discrete random variable with probability function 7 ? ~ % ~ % for % ~ What is the probability that ? is even? A) B) C) D) E)
2. For a certain discrete random variable on the non-negative integers, the probability
ACT245H1 S - PROBLEM SET 5
1. A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and without replacement. Calculate the probability that number of boys selected exceeds the number of girls selected. A) B) B) B)
B)
PROBLEM SET 9
Loss Distributions and Insurance 1. An insurer models the claim random variable ? for a certain insurance policy as follows: % 7 ? ~ % The insurer wishes to summarize the claim amount distribution with the parameters: ~ probability a non-zer