Problem 2 please. A car panel is spray-painted by a machine, and the technicians

Question

Problem 2 please. A car panel is spray-painted by a machine, and the technicians are particularly interested in the thickness of the resulting paint layer. Suppose that the random variable X measures the thickness of the paint in millimeters at a randomly chosen point on randomly chosen car panel, and that X takes value between 0.125 and 0.5 mm with a probability density function of f(x) = A[0.5 - (x - 0.25)^2] for 0.125 lessthanorequalto x lessthanorequalto 0.5 and f(x) = 0 elsewhere. a) Find the value of A that make f(x) valid pdf. b) Construct the cumulative distribution function. c) Find the probability that the paint thickness at particular point is less than 0.2 mm, larger than 0.35mm, and between 0.35 and 0.2 mm respectively. d) What is the expected paint thickness? e) What is the variance and standard deviation of point thickness? Repeating rolling die is considered as Bernoulli trials. A far die is rolled eight time. Calculate the probability that there are: a) Exactly free even numbers. b) Exactly one 6. c) No 4s. d) At least six prime numbers. On average there are four traffic accidents in a city during one hour of rush-hour traffic. Use the Poisson distribution to calculate the probability that is one such hour there are a) No accidents b) At least six accidents. The amount of sugar contained in 1-kg is actually normally distributed with a mean of mu = 1.03 kg and a standard deviation of sigma = 0.014 kg. a) What proportion of sugar packets are underweight? b) If an alternative package-filing machine is used for which the weights of the packets are normally distributed with a mean of mu = 1.05 kg and a standard deviation of sigma = 0.016 kg, does this result in an increase or a decrease in the proportion of underweight packets? The resistance of one motor of copper cable at a certain temperature is normally distributed with mean mu = 23.8 and variance sigma^2 = 1.28. a) What is the probability that a one-meter segment of copper cable has a resistance less than 23.0? b) What is the probability that a one-meter segment of copper cable has a resistance greater than 24.0? c) What is the probability that a one-meter segment of copper cable has a resistance between 24.2 and 24.5? Suppose that a fair coin is in times. Estimate the probability that the proportion of heads obtained lies between 0.49 and 0.51 for n = 100, 200, 500 1000, and 2000.

Explanation / Answer

a) Exactly five even numbers.

n = 8, p = probability of getting an even number in a single toss = 3/6 = 1/2, q = 1 - p = 1/2, x = 5

P(x) = C(n, x) p^x q^(n - x)

P(5) = C(8, 5) (0.5^5) 0.5^(8 - 5) = 0.2188

(b) Exactly one 6.

n = 8, p = probability of getting a 6 in a single toss = 1/6, q = 1 - p = 5/6, x = 1

P(x) = C(n, x) p^x q^(n - x)

P(1) = C(8, 1) (1/6)^1 (5/6)^(8 - 1) = 0.3721

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