2.1 A cellular call is made to your cell phone at random within a ten-minute interval. You were busy on a call for 2 minutes into this ten-minute period. There is a probability that the call arrived when you were not busy on your cell. Identify the random variable and define the distribution of the random variable?

2.2 Complete the below statement.

Identify the distribution of the random variable with the following moment-generating function. ?(?)=e6?+9?2

2.4 A soft drink giant, the Coca-Cola Company produces soft drinks in bulk. The amount of beverages sold to the retail chain stores (such as Pick 'n Pay, Checkers and Spar etc.) in one day can be modelled by a gamma or exponential distribution with ?? = 5 (measurements in litres).

2.4.1 Find the probability that the company will sell more than 5 kilolitres on a given day.

2.4.2 Interpret the answer found in (2.4.1).

2.5 There is relationship between the distribution function of a Beta-distributed random

variable and sums of binomial probabilities. Define this relationship.

2.6 The weekly repair cost ?? for a manufacturing machine has a probability density function given

by

?(?)= 4(1??)3,0?

0,elsewhere

with measurements in hundreds of rands. What type of distribution does Y follow?

Neyman-Pearson Lemma. (Lema Neyman-Pearson.) [2 marks] (b) Assume that scores of the students, Y for a mathematics course are normally distributed with unknown mean and variance, 30. We wish to test null hypothesis Hail = 75 against alternative hypothesis H,: 1 = 77. Let C = [Y : Y > 76.8) be critical region of the test, where Y is the observed mean sample. Compute the power of test if n = 50. [6 marks ] c) Let X,, X,..... X denote a random sample of size n from a Poisson distribution with mean A. Obtain the likelihood ratio critical region for testing Ho: 1 = 2 vs Hid = 5. [5 marks] "t X X,.....X, be a random sample from a normal distribution with unknown mean, and variance, o' = 0.16. We wish to test Ho : 0 = 5.0 versus H : 0 = 5.5.Q.1: a) Given a random sample of size n from a normal population with / = 0, use the Neyman-Pearson lemma to construct the most powerful critical region of size a to test the null hypothesis against the alternative b) Given a random sample of size n from a normal population with unknown mean and variance, find an expression for the likelihood ratio statistic for testing the null hypothesis against the alternative3-6. Hypothesis testing/The Neyman-Pearson Lemma and UMP Tests: The distribution for X under H0 and Ha is given below. x x) I Ho x) I Ha 1 0 0.05 2 0.005 0.015 3 0.005 0.002 4 0.005 0.045 5 0.005 0.005 6 0.01 0.08 7 0.01 0,05 8 0.01 0.001 9 0.01 0.03 10 0.01 0.035 11 0.015 0.12 12 0.015 005 5a) Determine the best critical region for an 0: = .05 test. 5b) What is the power of this test? 6a) If we decided that we want the rejection region to be x E {1 1,11,13,14}, what would our significance level be? 6b) What would be the power of our test in part 63? \"mam v The following statement forms part of the Neyman-Pearson lemma for a test of sizea of the hypotheses Ho versus Hi about a single parameter 0: LR = L(01 ; X) L(60; X) 2 A . (a) State the null and alternative hypotheses referred to above. [2 marks]~ (b) What desirable property does the test obtained using the Neyman- Pearsonlemma possess in this case? [2 marks]~ (c) Consider the following likelihood function for a single parameter 0 based on *, the mean of a sample of n observations from a particular distribution.~ L(0;X) = c . exp(-n0) Onx where c is a fixed but unknown constant.