For the given probability density function, over the given interval, find the mean, the variance, and the standard deviation. \[ f(x)=4,[2.25,2.50] \] A. $\mu=2.500 ; \sigma^{2}=0.0048 ; \sigma=0.069$ B. $\mu=2.375 ; \sigma^{2}=0.021 ; \sigma=0.144$ C. $\mu=2.375 ; \sigma^{2}=0.0052 ; \sigma=0.072$ D. $\mu=2.500 ; \sigma^{2}=0.0049 ; \sigma=0.070$
Solution
Step 1 ：The given probability density function is a constant function \(f(x) = 4\) over the interval [2.25, 2.50]. This is a uniform distribution.
Step 2 ：The mean (or expected value) of a probability density function (pdf) is given by the integral of \(x*f(x)\) over the given interval. For a uniform distribution, the mean is simply the midpoint of the interval. So, \(\mu = 2.375\).
Step 3 ：The variance is given by the integral of \((x - \mu)^2 * f(x)\) over the given interval. Calculating this gives \(\sigma^{2} = 0.0052\).
Step 4 ：The standard deviation is the square root of the variance. So, \(\sigma = \sqrt{0.0052} = 0.072\).
Step 5 ：Final Answer: The mean is \(\boxed{2.375}\), the variance is \(\boxed{0.0052}\), and the standard deviation is \(\boxed{0.072}\). Therefore, the correct answer is C. \(\mu=2.375 ; \sigma^{2}=0.0052 ; \sigma=0.072\).
