Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: $\mathrm { P } ( 2.5 \leq x \leq a ) = 0.5$
A)3
B)0.8
C)4
D)4.5

Question

Quizplus · Accepted Answer

For a uniform distribution from c to d, the probability density function is $f(x) = \frac{1}{d-c}$ for $c \leq x \leq d$. Here, $c = 2$ and $d = 6$, so $f(x) = \frac{1}{4}$. The probability that $x$ is between two values is the area under the curve between those values, which is simply the base (the difference between the two values) times the height (the probability density, $f(x)$). For $P(2.5 \leq x \leq a) = 0.5$, the base is $a - 2.5$, and the height is $f(x) = \frac{1}{4}$, so we have $\frac{1}{4}(a - 2.5) = 0.5$. Solving for $a$, we get $a - 2.5 = 2$, so $a = 4.5$.

Suppose X Is a Random Variable Best Described by a Uniform