Statistics 259 – Probability I
Midterm Exam Two
统计概率代写 1. Non-programmable, non-graphing calculators are permitted. 2. Check that your test paper has no missing, blank, or illegible pages.
1. Non-programmable, non-graphing calculators are permitted.
2. Check that your test paper has no missing, blank, or illegible pages.
3. Show all your work. Insufficient justification will result in a loss of marks.
4. Carry out your calculations to only 3 significant digits only.
Part I: Multiple Choice Questions: 统计概率代写
• There is only ONE correct answer in each question.
• For questions 1 to 5, 1 mark for choosing the right answer.
• For questions 6 to 10, 1 mark for choosing the right answer and 1 mark for explanation or/and calculation.
• Answer all of your multiple choice questions in the provided area.
Suppose that X is a discrete random variable that follows a binomial distribution with n trials and probability of success, p. When n is very large and p is very small which one of the following distributions will be a good approximation of the distribution of X?
(a) Negative binomial
Which one of the following statements does not apply to the of Bernoulli trials?
(a) If the trials are run for n times, the probability of success for the nth trial is determined by the probability of success for the (n − 1)th trial.
(b) In a sequence of Bernoulli trials, the probability of success in one trial is not determined by the probability of success of the other trials.
(c) There are exact two possible outcomes in a Bernoulli trial.
(d) A sequence of Bernoulli trials are independent repeated trials.
Suppose that X is a continuous random variable. Which one of the statement is NOT true.
(a) P(X = x) = 0
(b) The probability density function of X is the probability of X ∈ (a, b).
(c) The cumulative distribution function of X is a continuous function.
(d) The probability density function is the first derivative of the the cumulative distribution function.
Which one of the following statements is incorrect?
(a) If median of the distribution of a random variable, X, is 1/2, then P(X > 1/2) = 1/2
(b) Suppose that X is a discrete random variable, then the probability mass function of X is the probability of X equals to a value.
(c) Suppose that X is a continuous random variable, then the probability density function of X is the probability of X equals to a value.
(d) If X is a real valued random variable, then P(−∞ < X, ∞) = 1.
If the probabilities of having a female child are 0.5, then the probability that a family’s seventh child is their second daughter is:
A shipment of 20 digital voice recorders contains 5 that are defective. What is the expected numbers of defects in this shipment?
Part II – Written Questions 统计概率代写
The probability distribution of a discrete random variable X is
(a) Find P(X < 3).
(b) Find E(X).
(c) Find V ar(3 + 2√X)
Suppose that the percentage of drivers who are multitaskers is approximately 60%. In a random sample of 20 drivers, let X be the number of multitaskers.
(a) Whatis the distribution of X? Justify your answer.
(b) What is the mean (µ), variance (σ2 ) and standard deviation (σ) of X?
(c) What is the probability that at least 7 drivers in the sample are multitaskers?
An Internet-based company that sells discount accessories for cell phones often ships an excessive number of defective products. The company needs better control of quality. Suppose it has 20 identical car chargers on hand but that 5 are defective.
(a) Let the number of defective chargers be a random variable, Y , What is the distribution of Y ?
(b) If the company decides to randomly select (without replacement) 10 of these items, what is the probability that 2 of the 10 will be defective?
(c) Find the mean of the probability distribution of the number of defectives in a sample of 10 randomly chosen for inspection.
(d) Find the standard deviation of the probability distribution of the number of defectives in a sample of 10 randomly chosen for inspection.
In large city, telephone calls (Y ) to EMS follows a Poisson distribution. Suppose that on average, six calls will arrive during 30-minute period.
(a) What a is the probability that 3 calls arrive in a 30-minute period? [2 marks]
(b) What is the probability of no more than three calls arriving in a 30-minute period? [2 marks]
(c) Suppose that the cost (in dollars) to respond to Y calls is given by
C = 150 + 100Y
Find the expected value and variance of C.
(a) Derive the mgf of Y
(b) Use the mgf, fifind expressions for E(Y ) and V ar(Y ).