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Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function.

f(x) = 2(x+2)/5, 0

User Franey
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1 Answer

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Final answer:

The question is about finding the probability distribution, graphing it, and calculating probabilities and percentiles based on the provided probability density function. Since the function is only defined from 0 to 1, probabilities for values outside this range are zero.

Step-by-step explanation:

The student is asking a question that involves calculating various probabilities given a probability density function (PDF) for a random variable X which represents the proportion of individuals responding to a mail-order solicitation. The question includes tasks such as graphing the distribution, finding the probability that X falls between two values and finding a specified percentile.

State the probability density function:

f(x) = 2(x+2)/5 for 0 < x < 1 and f(x) = 0 otherwise.

Graph the distribution:

Create a graph where the horizontal axis represents X and the vertical axis represents f(x). The graph will show a linearly increasing line from f(0) = 0 to f(1) = 4/5, and then it will drop back to 0 after X=1.

Find P(2 < x < 10):

This is actually a trick question since the support of the PDF is only from 0 to 1. Therefore, P(2 < x < 10) = 0.

Find P(x > 6):

Similarly, since the support of f(x) is from 0 to 1, P(x > 6) = 0.

Find the 70th percentile:

To find the 70th percentile, you need to calculate the cumulative distribution function (CDF) and then solve for the X value that gives a CDF of 0.7. This would typically involve integrating the PDF and solving the resulting equation, but since this function is linear and only defined on [0,1], we can assume if it were correctly given, it would involve an easier calculation, often solvable by algebraic manipulation.

User Djpohly
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