If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$

Question

I'm stuck on this question, can someone help me, many thanks.

If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$?

update(this problem has been solved): I made a mistake when calculating, the result should be under the condition : x ranges from a to b. The final result is 1/3.

Answer 1

Hint:- Let us suppose $U$ follows Uniform distribution with parameter $a$ and $b$.

Mean=$E(U)=\frac{b+a}{2}=5$ and Variance $=V(U)=\frac{(b-a)^2}{12}=3$.

Answer 2

As the mean is directly in the middle between $a$ and $b$ you can set

• $a = 5-x$ and $b=5+x$ for $x>0$

Now solve $$\sigma^2 = \frac{(b-a)^2}{12}= \frac{(2x)^2}{12}=\frac{x^2}{3} = 3\stackrel{x>0}{\Rightarrow} x= 3$$ So, $U$ "lives" on $[a,b]=[2,8]$. It follows $$P(U<4) = \frac{4-2}{8-2}=\frac{2}{6}=\frac{1}{3}$$