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

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.

• uniformly distributed over what set? – mathworker21 Jan 17 at 12:44
• Where did you get stuck with solving this? What have you tried already? Questions without context and visible effort often get downvoted and closed. – postmortes Jan 17 at 12:58

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

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

• You are assuming that $U$ is continuous Uniform. All that is stated is that it is Uniform (which might be continuous or discrete) – wolfies Jan 17 at 12:47
• Yes I know this formula, the result is 1/3 right? I just realized I made a mistake, x should be in the range from a to b – wawawa Jan 17 at 12:48
• @ Cecilia Yes, you are correct. I forgot to mention one thing that you solve the two equations under the restriction that $a<b$. – user440191 Jan 17 at 12:50
• @ wolfies you are absolutely right. I should have mentioned that I am assuming continuous uniform distribution. However, if one assume that it is the case of discrete uniform, I think thee will be a issue regarding the consideration of the set over which it is uniformly distributed ( as mentioned by @mathworker above) – user440191 Jan 17 at 12:58
• I urge @Cecilia to consider the discrete case when the set is $\{1,2,\dots N\}$ and see what happens. – user440191 Jan 17 at 12:59

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 $$P(U<4) = \frac{4-2}{8-2}=\frac{2}{6}=\frac{1}{3}$$

• Thank you so much for the answer, however @Bhargob answers the question before you haha and I've already chosen his answer, thanks for your time and I really appreciated it. – wawawa Jan 17 at 14:59