Independent solution

How to solve this Uniform Distribution question

Answer in brief

For a continuous uniform distribution, both the mean and median are the interval midpoint, while the 75th percentile is one-quarter of the lower endpoint plus three-quarters of the upper endpoint. Their sum is (5a + 7b)/4, which is choice B.

Setup

Setup

Use the standard mean and quantile formulas for a continuous uniform random variable on an interval with endpoints a and b.

X∼Uniform⁡(a,b),qp=a+p(b−a)X\sim\operatorname{Uniform}(a,b),\qquad q_p=a+p(b-a)

Model

Model

The mean and median coincide at the midpoint, and the upper quartile follows from the quantile formula.

x=a+b2,y=a+b2x=\frac{a+b}{2},\qquad y=\frac{a+b}{2}
z=q0.75=a+34(b−a)=a+3b4z=q_{0.75}=a+\frac{3}{4}(b-a)=\frac{a+3b}{4}

Compute

Compute

Add the three location measures using a common denominator.

x+y+z=(a+b)+a+3b4x+y+z=(a+b)+\frac{a+3b}{4}
x+y+z=5a+7b4x+y+z=\frac{5a+7b}{4}

Answer

Answer

The required symbolic sum is the expression listed under choice B.

5a+7b4(B)\boxed{\frac{5a+7b}{4}\quad\text{(B)}}