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.