Download Note: Numerical result must correctly give at least 4 significant digits

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript 
Note: Numerical result must correctly give at least 4 significant digits.
1. If A, B, C and D are mutually independent, and Pr(A) = 0.23, Pr(B) =
0.42, Pr(C) = 0.54 and Pr(D) = 0.81, find
Pr[(A ∩ C̄) ∪ (B ∩ D) ∪ (A ∩ D̄)]
2. Let X and Y have a bivariate distribution with the following joint probability density function:
5
2
when 0 < y < 1 and 0 < x < 1 − y 2
2 (x + y )
f(x, y) =
0
otherwise
Find:
(a) The marginal probability density function of each X and Y ,
(b) Cov(X, Y ),
(c) the conditional probability density function of Y given that X = 12 .
3. An integer-valued random variable X has the following probability generating function
−4
P (z) = 2 − exp(z − 1)
Find
(a) its mean and standard deviation,
(b) Pr(X ≥ 5).
(c) Pr(X̄ ≥ 5), where X̄ is a sample mean of 6 independent observations
from this distribution.
4. Consider the following experiment: A die is rolled 30 times, followed by
flipping a coin as many time as the total number of sixes obtained in the
first part of the experiment.
(a) Using composition of the corresponding PGFs, find the expected
value and variance of the number of heads one will thus get,
(b) and the probability that this number will be between 5 and 10 (inclusive).
5. Using convolution, find the PDF of the sum of three independent random
variables from the following distribution:
(a) Normal, with the mean of 0 and the variance of 1,
(b) Exponential, with the mean of 1,
(c) Cauchy, with the median of 0 and quartile deviation of 1.
Identify each answer.
1
Related documents