Final answer:
The probability density function is f(x) = 1 from 5.8 to 6.8 years. The mean (μ) is 6.3 years, and the standard deviation (σ) is approximately 0.289 years. The probability P(6 < X < 6.5) is 0.5, and the distribution is symmetric.
Step-by-step explanation:
Answer to Student's Question on Uniform Distribution
a. The probability density function (PDF) for this uniform distribution, where the random variable X represents the age of a first grader, is given by:
f(x) = 1 / (6.8 - 5.8) = 1 / 1 = 1 for 5.8 ≤ x ≤ 6.8
b. To calculate the mean (μ) and standard deviation (σ), we use the formulas for a uniform distribution. The mean is (a + b) / 2 and the standard deviation is √((b - a)^2 / 12), where a and b are the minimum and maximum values of the age range, respectively.
μ = (5.8 + 6.8) / 2 = 6.3 years
σ = √((6.8 - 5.8)^2 / 12) = √((1)^2 / 12) = √(1 / 12) = √(0.0833) ≈ 0.289 years
c. To find P(6 < X < 6.5), we calculate the area under the PDF between 6 and 6.5, which in a uniform distribution is simply:
P(6 < X < 6.5) = (6.5 - 6) / (6.8 - 5.8) = 0.5 / 1 = 0.5
d. The age distribution is symmetric because the PDF for a uniform distribution is constant across the interval [a, b]. Thus, the distribution has a rectangular shape and is symmetric about the mean.