Question

A random variable X follows the continuous uniform distribution with a lower bound of −2 and an upper bound of 16. 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.)

Answer 1

Answer:

a. Height=0.0555

b. Mean=7

SD=4.0415

c. P(x≤1)=0.1665

Step-by-step explanation:

a. If we have a continous uniform distribution, we have a density function that is a constant:

$f(x)=C$

In the other hand, the cumulative density function across the entire region should be one:

$F(x)=\int\limits^{-2}_{16} {f(x)} \, dx =\int\limits^{-2}_{16} {C} \, dx=1\\\\C(16-(-2))=1\\\\C*18=1\\\\C=1/18=0.0555$

Then the height C of the density function is 0.0555.

b. The mean of this distribution is:

$M=\frac{1}{2}(a+b)=\frac{1}{2}(-2+16) =\frac{1}{2}* 14=7$

The standard deviation of this distribution is:

$S.D.=\frac{a+b}{\sqrt{12}} =\frac{-2+16}{\sqrt{12}} =4.0415$

c.

$P(x\leq1)=0.0555*\int\limits^1_{-2} {} \, dx =0.0555*(1-(-2))=0.0555*3=0.1665$