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Based on past experience, it is assumed that the number of flaws of per metre in rolls of wrapping paper follows a Poisson distribution with a mean of 2 flaws per 4 metres of paper. The probability (correct to 2 decimal places) that more than 2 flaws will be observed in 5 metres of wrapping paper produced is:
A random sample of 11 observations was taken from normal population. The sample mean and
standard deviation are 74.5 and 9 accordingly. Can we infer at 5% significance level that the
population mean is greater than 70?
5. Repeat number 4 with assuming the population standard deviation = 9
In a survey conducted among a random sample of students the following observations were made regarding their gender and learning environment preferences during the COVID-19 pandemic:
168 prefer online learning
202 prefer face to face learning
180 prefer blended learning
34 male students prefer online learning and
70 male students prefer blended learning
106 female students prefer face to face learning
Required:
a) What is the probability that a female student is chosen?
b) What is the probability that a male student prefers face to face learning?
c) What is the probability that a student prefers online or blended learning?
d) If it’s known that the student is female, what is the probability that this student prefers online learning.
e) Using a practical example, explain the difference between mutually exclusive events and independent events.
The following are the loss amount in thousands of dollars from three portfolio of insurance policies
Portfolio Y1: 43 30 27 24
Portfolio Y2: 43 30 27 24
Portfolio Y3: 43 30 27 24
Portfolio Y4: 43 30 27 24
Determine
i) The mean vectors of the portfolios
ii)Variance covariance matrix of the portfolios
iii) Correlation matrix of the portfolios
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
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?
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.
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