Hi there, I have an exam on the 17th May at 12:00 Midday (GMT). It is a 2 hour probability...

Question

Hi there, I have an exam on the 17th May at 12:00 Midday (GMT). It is a 2 hour probability (University Level) exam and was wondering if TAE provides a service which can solve the answers within a 2 hour period.Perfect, the layout of the exam is to answer 3 out of the 4 questions. We have 2 hours to do the exam. On the day the exam questions are released, I can send these over to TAE and will they be able to get the answers done (with working out) within an hour and a half?I have uploaded exam questions from 3 papers.AM10PB Probability - Mock Examination 1 1. Let (Ω,F ,P) be a probability space and {B1, B2, . . . } be a partition of Ω with P(Bi) > 0 ∀ i. (a) State the definition for F to be an event space. (3 marks) (b) if A,B ∈ F , prove that A ∩B ∈ F . (3 marks) (c) State the definition of the conditional probability P(A|B). (3 marks) (d) State and prove the partition theorem for computing P(A). (5 marks) Suppose two generators provide electrical power for a pneumatic drill. Each generator is independent of the other, and each work with prob- abilities 0.4 and 0.5 respectively. If both generators work, then enough power is provided with certainty. If only one is working, then there is only enough power with probability equal to 0.6. If neither work then there is not enough power. (e) What is the probability of there being enough power for the pneu- matic drill? (6 marks) (Total: 20 marks) This examination is subject to the Examination Regulations for Candidates AM10PB Probability - Mock Examination 2 2. You have a large bag of 100 coloured sweets. There are 15 blue, 15 yellow, 30 red and 40 green sweets. Your favourite colour is blue. (a) Let X be a discrete random variable with a binomial distribution with parameters n and p. Write down the probability mass function fX of X. (3 marks) (b) What is the probability of selecting a blue sweet from the bag? (2 marks) (c) Let the value of X be the number of blue sweets. Given that you pick 10 sweets randomly with replacement. What is the expected number of blue sweets you would have picked? (3 marks) (d) Still assuming that after each selection you place the sweet back inside the bag. What is the probability of picking 3 blue sweets in 10 picks? (4 marks) (e) You decide now to eat each sweet once picked. Given that you eat 10 sweets in a row, what is the probability that you have eaten 4 blue sweets? (4 marks) (f) What is the probability that you have eaten a blue sweet? (4 marks) (Total: 20 marks) This examination is subject to the Examination Regulations for Candidates AM10PB Probability - Mock Examination 3 3. Let X be a discrete random variable with mass function fX given by X 0 1 2 3 4 5 fX c 2c 3c 0 c c (a) State the definition of X being a discrete random variable? (4 marks) (b) Define the expectation of the quantity g(X) in terms of fX . (2 marks) (c) What is the image of the above discrete random variable X? (2 marks) (d) Determine the constant c for which fX is a probability mass func- tion. (3 marks) (e) Compute the expectation of the discrete random variable X. (3 marks) (f) Compute the variance of the discrete random variable X. (3 marks) (g) Compute the probability of the discrete random variable X taking odd values. (3 marks) (Total: 20 marks) This examination is subject to the Examination Regulations for Candidates AM10PB Probability - Mock Examination 4 4. Let X be a normal distributed continuous random variable with param- eters µ and σ2 such that X ∼ N(µ, σ2). (a) State the probability density function fX of X (3 marks) Let Φ(z) = P(Z ≤ z) be the standard normal cumulative distribution function of the standard normal variable Z ∼ N(0, 1). (b) Let µ = 4 and σ = 3, compute P(X > 5) given that Φ(1/3) = 0.6306. (5 marks) (c) Assuming Z is a standard normal distributed variable, show that E[exp(2Z)] = exp(2). (3 marks) Two jointly distributed continuous random variables X and Y have joint density function fX,Y (x, y) = { c(x+ 3y)e−(x+y) for 0 ≤ x  X) = 5/8. (5 marks) (Total: 20 marks) END OF EXAMINATION PAPER This examination is subject to the Examination Regulations for Candidates   AM10PB 1. Let (Ω,F ,P) be a probability space. (a) State the definition for P : F → [0, 1] to be a probability measure. (3 marks) (b) Prove that the conditional probability measure Q(A) = P(A|B) is a probability measure given that P(B) > 0. (5 marks) (c) Given two events A,B ∈ F , prove that P(A) = P(A ∩B) + P(A ∩Bc). (4 marks) There is an 80% chance that it will rain today, while there is only a 50% chance that it will be raining tomorrow. The chance that it will rain on both days is 60%. (d) Determine the probability that it is raining at least once in the next two days. (4 marks) (e) Determine the probability that it rains tomorrow given that it rains today. (2 marks) (f) Is the event that it rains today dependent on if it is raining tomor- row? Justify your answer. (2 marks) (Total: 20 marks) 1 of 4 This assessment is subject to the University Assessment Regulations for Candidates AM10PB 2. A food manufacturer produces 1,000 boxes of cereal each containing a toy. However, there was a packaging error and it is expected that 100 of these boxes have no toy. A sample of 20 boxes are opened. Let X be a discrete random variable whose value represents the number of cereal boxes without a toy. (a) Write down the distribution of the random variable X. Make sure to include numerical values for each parameter. (3 marks) (b) Calculate the probability that all 20 boxes will have a toy. (4 marks) (c) Calculate the probability that fewer than 3 boxes in the sample will not have a toy. (4 marks) Let (Ω,F ,P) be a probability space. Given that {B1, B2, . . . } forms a partition of a sample space Ω and that P(Bi) > 0 ∀ i. Then for any event A ∈ F with P(A) > 0, Bayes’ theorem states P(Bj|A) = P(A|Bj)P(Bj)∑∞ i=1 P(A|Bi)P(Bi) . (d) Prove Bayes’ theorem. (5 marks) There are three bags, each containing a set of balls. Bag A has 3 white and 2 black balls, Bag B has 3 white balls and 1 black ball; and Bag C has 3 white and 4 black balls. A bag is selected at random and a ball is drawn and is found to be white. (e) Calculate the probability that Bag C was selected. (4 marks) (Total: 20 marks) 2 of 4 This assessment is subject to the University Assessment Regulations for Candidates AM10PB 3. Let X be a normally distributed continuous random variable with pa- rameters µ and σ2 such that X ∼ N(µ, σ2). (a) Write the probability density function fX(x) of X. (3 marks) (b) State the transformation from X ∼ N(µ, σ2) to the standard nor- mal variable Z ∼ N(0, 1). (2 marks) Let Φ(z) = P(Z ≤ z) be the standard normal cumulative distribution function (CDF) of the standard normal variable Z ∼ N(0, 1). (c) Let µ = 2 and σ2 = 9. Compute the probability P(2 ≤ X ≤ 3) giving your answer in terms of the standard normal CDF Φ(z). (5 marks) Let a continuous random variable Y have probability density function given by fY (y) = { c exp(3− 4y) for 0 ≤ y

Rajeswari · Accepted Answer

The event space F is the set of all distinct possible outcomes of the random process. Event space sometimes called sample space contains distinct outcomes and also these exhaust all the possibilities
If we write prob for all elements in the event space, the sum of all probabilities would always be equal to 1.
A and B are two possible outcomes in event space F. A\B, the symmetric difference of A and B is nothing but all outcomes in event A but not in B.  In Venn diagram this would be denoted as A-B or A-A intersectin B
Since A and B are in the set F, A-B being a proper subset of A also must belong to F
Standard event space F would be sets A,B,C,D… which are distinct and disjoint.  These if grouped would be exhaustive of F
F={A,B,,..} which are disjoint mutually exclusive but exhaustive of F
For example, when we toss a die, the events are 
{1,2,3,4,5,6} the outcomes and form a std event space
A – it snows on Monday
B – it snows on Tuesday
Prob that it snows atleast on Monday or Tuesday 
=P(AUB) = P(A)+P(B)-P(A⊓B)
= 0.75+0.50-0.40 = 0.80
P(A⊓B)=0.40
P(A)*P(B) = 0.75*0.50 = 0.

Hi there, I have an exam on the 17th May at 12:00 Midday (GMT). It is a 2 hour probability (University Level) exam and was wondering if TAE provides a service which can solve the answers within a 2...

Answer To: Hi there, I have an exam on the 17th May at 12:00 Midday (GMT). It is a 2 hour probability...

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment