Question

Given that X is uniformly distributed over the interval (7,18):

Provide a solution showing your calculations and submit your work for marking.

a. Calculate P(10 < X < 13).

Include a sketch as part of your complete solution.

P(10 < X < 13) =

Round to three decimal places if necessary

b. Determine the mean (µ) and standard deviation (σ) of the distribution.

µ =

Round to three decimal places if necessary

σ =

Round to three decimal places if necessary

Answer 1

To calculate P(10 < X < 13), integrate the uniform distribution's pdf from 10 to 13, resulting in approximately 0.273. The mean (µ) and standard deviation (σ) of X are 12.5 and approximately 3.162, respectively, calculated using the uniform distribution formulas.

Step-by-step explanation:

Calculating Probability for a Uniform Distribution:

Given that X is uniformly distributed over the interval (7,18), to calculate P(10 < X < 13), we recognize that the probability density function (pdf) of a uniform distribution is 1/(b-a) where 'a' and 'b' are the endpoints of the interval. Here, a = 7 and b = 18, so the pdf is 1/(18-7) = 1/11.

The probability P(10 < X < 13) is found by integrating the pdf from 10 to 13, which gives us the area under the probability density function curve between these two points. Thus, P(10 < X < 13) = (13 - 10) * (1/11) = 3/11 ≈ 0.273.

To determine the mean (µ) and standard deviation (σ) of the distribution, we use the formulas for a uniform distribution: μ = (a + b) / 2 and σ = √((b - a)^2 / 12). Therefore, µ = (7 + 18) / 2 = 12.5 and σ = √((18 - 7)^2 / 12) ≈ 3.162.