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.)
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. The height of the density function f(x) for a continuous uniform distribution is 1 / (upper bound - lower bound). In this case, the upper bound is 12 and the lower bound is -2:
f(x) = 1 / (12 - (-2))
= 1 / 14
= 0.0714 (rounded to 4 decimal places)
So, the height of the density function is 0.0714.
b. The mean (μ) and standard deviation (σ) for a continuous uniform distribution can be calculated using the following formulas:
mean (μ) = (lower bound + upper bound) / 2
standard deviation (σ) = (upper bound - lower bound) / √12
In this case, the lower bound is -2 and the upper bound is 12:
mean (μ) = (-2 + 12) / 2
= 10 / 2
= 5
standard deviation (σ) = (12 - (-2)) / √12
= 14 / √12
= 4.0416 (rounded to 2 decimal places)
So, the mean is 5 and the standard deviation is 4.04.
c. To calculate P(X ≤ -1), we need to find the cumulative distribution function (CDF) at -1.
CDF(x) = (x - lower bound) / (upper bound - lower bound)
In this case:
CDF(-1) = (-1 - (-2)) / (12 - (-2))
= 1 / 14
= 0.0714 (rounded to 4 decimal places)
So, P(X ≤ -1) is 0.0714.
To answer these questions, we need to apply the formulas and concepts related to the continuous uniform distribution.
a. The height of the density function, f(x), for a continuous uniform distribution is given by the formula:
f(x) = 1 / (b - a)
where 'a' is the lower bound and 'b' is the upper bound of the distribution.
In this case, the lower bound (a) is -2 and the upper bound (b) is 12. Substituting these values into the formula:
f(x) = 1 / (12 - (-2)) = 1 / 14
Therefore, the height of the density function f(x) is 1/14, which is approximately 0.0714 when rounded to 4 decimal places.
b. The mean (μ) and the standard deviation (σ) for a continuous uniform distribution are given by the following formulas:
Mean (μ) = (a + b) / 2
Standard Deviation (σ) = sqrt((b - a)^2 / 12)
In this case, the lower bound (a) is -2 and the upper bound (b) is 12. Substituting these values into the formulas:
Mean (μ) = (-2 + 12) / 2 = 10 / 2 = 5
Standard Deviation (σ) = sqrt((12 - (-2))^2 / 12) = sqrt(14^2 / 12) = sqrt(196 / 12) ≈ sqrt(16.33) ≈ 4.04
Therefore, the mean of the distribution is 5, and the standard deviation is approximately 4.04 when rounded to 2 decimal places.
c. To calculate P(X ≤ -1) for a continuous uniform distribution, we need to integrate the density function f(x) from the lower bound (-2) to -1.
The formula to calculate the probability of a continuous uniform distribution is given by:
P(X ≤ x) = (x - a) / (b - a)
where 'x' is the upper limit of the integration and 'a' and 'b' are the lower and upper bounds of the distribution, respectively.
In this case, substituting the values into the formula:
P(X ≤ -1) = (-1 - (-2)) / (12 - (-2)) = 1 / 14
Therefore, P(X ≤ -1) is approximately 0.0714 when rounded to 4 decimal places.