Question

In: Statistics and Probability

A worn machine is known to produce 10% defective components. If the random variable X is...

A worn machine is known to produce 10% defective components. If the random variable X is the number of defective components produced in a run pf 3 components, find the probabilities that X takes the values 0 to 3.

Suppose now that a similar machine which is known to produce 1% defective components is used for a production run of 40 components.We wish to calculate the probability that two defective items are produced. Essentially we are assuming thatX~B(40,0.01) and we use both the binomial distribution and its Poisson approxiamation for comparison.

Solutions

Expert Solution

Solution

Assuming that the production of components is independent and that the probability p = 0.1 of producing a defective component remains constant, the following table summarizes the production run. We let G represent a good component and let D represent a defective component. Note that since we are only dealing with two possible outcomes, we can say that the probability q of the machine producing a good component is 1 − 0.1 = 0.9. More generally, we know that q+p = 1 if we are dealing with a binomial distribution.

From this table it is easy to see that

P(X = 0) = (0.9)3

P(X = 1) = 3 × (0.9)2 (0.1)

P(X = 2) = 3 × (0.9)(0.1)2

P(X = 3) = (0.1)3

Clearly, a pattern is developing. In fact you may have already realized that the probabilities we have found are just the terms of the expansion of the expression (0.9 + 0.1)3

since (0.9 + 0.1)3 = (0.9)3 + 3 × (0.9)2 (0.1) + 3 × (0.9)(0.1)2 + (0.1)3

Using the binomial distribution we have the solution

P(X = 2) = 40C2(0.99)40−2 (0.01)2 =(40*39)/(1*2)× 0.9938 × 0.012 = 0.0532

Note that the arithmetic involved is unwieldy. Using the Poisson approximation we have the solution

P(X = 2) = e −0.4 (0.42/2!)= 0.0536

Note that the arithmetic involved is simpler and the approximation is reasonable


Related Solutions

X is a Gaussian random variable with variance 9. It is known that the mean of...
X is a Gaussian random variable with variance 9. It is known that the mean of X is positive. It is also known that the probability P[X^2 > a] (using the standard Q-function notation) is given by P[X^2 > a] = Q(5) + Q(3). (a) [13 pts] Find the values of a and the mean of X (b) [12 pts] Find the probability P[X^4 -6X^2 > 27]
For a discrete random variable X, it is known that Pr[X = 1] = 0.15, Pr[X...
For a discrete random variable X, it is known that Pr[X = 1] = 0.15, Pr[X = 3] = 0.3, Pr [X = 5] = 0.35, and Pr[X = 7] = 0.2. Find Pr[X < 5].
6.2% of cable boxes of a certain type are defective. Let the random variable X represent...
6.2% of cable boxes of a certain type are defective. Let the random variable X represent the number of defective cable boxes among 200 randomly selected boxes of this type. Suppose you wish to find the probability that X is equal to 8. (i) Does the random variable X have a binomial or a Poisson distribution? How can you tell? (ii) If X has a binomial distribution, would it be reasonable to use the Poisson approximation? If not, why not?
The number of defective items produced by a machine (Y) is known to be linearly related...
The number of defective items produced by a machine (Y) is known to be linearly related to the speed setting of the machine (X). Data is provided below. a) (3) Fit a linear regression function by ordinary least squares; obtain the residuals and plot the residuals against X. What does the residual plot suggest? b) (3) Plot the absolute value of the residuals and the squared residuals vs. X. Which plot has a better line? c) (4) Perform a weighted...
Determine whether or not the random variable X is a binomial random variable. (a) X is...
Determine whether or not the random variable X is a binomial random variable. (a) X is the number of dots on the top face of a fair die (b) X is the number of hearts in a five card hand drawn (without replacement) from a well shuffled ordinary deck. (c) X is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which 0.02% of all parts are defective. (d) X...
1. Let X be a continuous random variable such that when x = 10, z =...
1. Let X be a continuous random variable such that when x = 10, z = 0.5. This z-score tells us that x = 10 is less than the mean of X. Select one: True False 2. If an economist wants to determine if there is evidence that the average household income in a community is different from $ 32,000, then a two-tailed hypothesis test should be used. Select one: True False 3. α (alpha) refers to the proportion of...
Of the parts produced by a particular machine, 1% are defective. If a random sample of...
Of the parts produced by a particular machine, 1% are defective. If a random sample of 8 parts produced by this machine contains 2 or more defective parts, the machine is shut down for repairs. Find the probability that the machine will be shut down for repairs based on this sampling plan.
Question Two: The cleaning time of a grinding machine is a random variable (X). The cleaning...
Question Two: The cleaning time of a grinding machine is a random variable (X). The cleaning time is usually measured in second. The maintenance department reported the following information for machines A, B, and C. A = {XA | x < 80.54} B = { XB | 20.35 < x < 40.23} C= { XC | x < 20.23} For future maintenance planning and scheduling, HELP in finding the following events: (a) A′ ∩ B (b) (A ∩ B) (c)...
A factory that puts together car parts is known to produce 3% defective cars. Following a...
A factory that puts together car parts is known to produce 3% defective cars. Following a fire outbreak at the factory, reconstruction is carried out which may result in a change in the percentage of defective cars produced. To investigate this possibility, a random sample of 200 cars is taken from the production and a count reveals 14 defective cars. What may be concluded? Run a significance test using a 0.05 α-level of significance.
A) Machine A produces an average of 10% of defective parts. Machine B produces on average...
A) Machine A produces an average of 10% of defective parts. Machine B produces on average twice as many defective parts as machine A. Machine C produces on average three times more defective parts than machine B. Here are 8 bags of parts from A, 2 bags of parts from B and 1 bag of coins from C. We randomly take a coin from one of the bags itself also taken at random. She is defective. 1. What is the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT