a. The height of the density function f(x) is given by 1 / (upper bound - lower bound), which in this case is 1 / (12 - (-2)) = 1/14. Therefore, the height of the density function is 0.0714.
b. The mean of a continuous uniform distribution is given by (upper bound + lower bound) / 2, which in this case is (12 + (-2)) / 2 = 5. The standard deviation is given by the formula (upper bound - lower bound) / sqrt(12), which in this case is (12 - (-2)) / sqrt(12) ≈ 3.27.
c. P(X ≤ -1) is the area under the density function curve to the left of -1. Since the distribution is uniform, this probability can be calculated as the relative width of the interval from the lower bound (-2) to -1, divided by the total width of the distribution (12 - (-2)). Therefore, P(X ≤ -1) = (1 - (-2)) / (12 - (-2)) = 3 / 14 ≈ 0.2143.
A random variable X follows the continuous uniform distribution with a lower bound of −2 and an upper bound of 12.
a. What is the height of the density function f(x)? (Round your answer to 4 decimal places.)
b. What are the mean and the standard deviation for the distribution? (Round your answers to 2 decimal places.)
c. Calculate P(X ≤ −1). (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)
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