For a branching process, calculate when
To solve this numerically, we will use the following general equation for the probability of extinction \(\pi_o\) in a branching process: \[ \pi_o = P_0 + P_1 \pi_o + P_2 \pi_o^2 + P_3 \pi_o^3 + \ldots \] This equation represents the extinction _______ __________ _________ _______ _________.
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For a branching process, calculate when
The joint probability mass function of X and Y , p ( x , y ) , is given by
i)
ii) Find P[x/y=1]
iii) Marginal distributions of X and Y
Twenty-five portfolio managers were evaluated in terms of their performance. Suppose Y represents the rate of return achieved over a period of time; Z1 is the manager’s attitude toward risk measured on a five-point scale from “very conservative” to “very risky”; and Z2 is years of experience in the investment business. The observed correlation coefficients between pairs of variables are
i) Interpret the sample correlation coefficients and
ii) Calculate the partial correlation coefficient and interpret this quantity with respect to the interpretation provided for
in Part (i)
Which of the following statements are true or false. Give reasons
a) is the quadratic form of a positive definite matrix
b) In a bivariate normal distribution then x and y are independent
c) The relation of accessibility in states is transitive.
d) For any pair of discrete random variables X and
e) If {X (t) : t 0 } is a Poisson process, then
, where a is a constant is also a Poisson process.
Suppose the joint density function of X and Y is given by
Let X have covariance matrix
i) Determine p and V 1/2
ii) Multiply your matrices to check the relation V 1/2 P V 1/2
Suppose that people arrive at a bus stop in accordance with a Poisson process with rate . The bus departs at time t . Let X denote the total amount of waiting time of all those that get on the bust at time t . We want to determine Var ( X ) . Let N (t ) denote the number of arrivals by time t
i) What is E [ X | N (t)] ?
ii) Argue that Var
iii) What is Var ( X ) ?