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State the various properties of covariance matrix of a random vector
Define a multivariate normal distribution with respect to a given linear combination
Define the following theorems with respect to a random vector
i) Central limit theorem
ii) Weak law of large numbers
Modulus of rigidity G=(R^4 θ)/L, where R is the radius, θ is the angle of twist and L length. Using Total Differential, determine the approximate change in G, and hence the approximate percentage change in, G when R is increased by 2%, θ is reduced by 5% and L is increased by 4%
inverse of 1 2 3
4 5 3
7 8 9
Question 2 [25] Suppose that the latest census indicates that for every 10 young people available to work only 4 are employed. Suppose a random sample of 20 young graduates is selected. Required: a) What is the probability that they are all employed? b) What is the probability that none of them are employed? c) What is the probability that at least four are employed? d) What is the probability that at most fifteen are employed? e) What is the probability that the number of young graduates who are employed is greater than ten but less than fifteen? f) What is the expected number of graduates who are not employed? g) What is the standard deviation for the number of graduates who are not employed?
Emmerson invests N$100,000.00 on a monthly basis, at the end of each month at an interest rate of8.5% compounded quarterly.
a) How much will he have after 3.5 years? (6)b) How much interest does he earn on a monthly basis? (Hint: Calculate the monthly average)
Assume that X has a uniform distribution over [0; 1] and that Y has the uniform distribution over
[2; 3]: Which of the following statements are true and which are false? Justify your answers!
(a) P (X < Y) = 1.
(b) Since X is smaller than Y , P (X < 1) > P (Y < 1).
(c) There are some values a for which P (X < a) = P (Y < a).
In each of the following scenarios, identify the distribution of the random variable X by giving the distribution and the parameter(s). Justify your answer!
(a) Suppose that drilling for water in a region is successful with probability 0.03 for each hole, independently of other holes. Assume that 10 holes are drilled. Let X be equal to one if at least one of the holes found water, and zero otherwise.
(b) A box contains 10 USB flash drives, and it is known that among them are five second hand flash drives which already contain data. Two flash drives are randomly taken from the box. X is equal to the number of flash drives chosen which already contain data.
(c) Assume that 1% of USB flash drives bought from a street corner vendor are known to be defective. One flash drive is bought from the vendor each day, and X is the number of defective flash drives bought during the first three days.
Assume that the times between arrivals of customers to a store are independent of each other and
exponentially distributed with an average of 10 minutes.
(a) Find the probability that the time between arrivals is more than 30 minutes.
(b) Find the probability that two customers arrive at the same time.
(c) Find the probability that the time until the next arrival is more than 30 minutes, given that it is
more than 20 minutes.
(d) If two customers arrived between 12:50 and 13:00 and one at 13:00, what is the probability
that the next customer arrives before 13:15?
#340153 on Dec 2023