How do you compute uniform distribution with only mean and proportion given?

Question

Machine A produces mints that have a label weight of 50g and is believed the weights of the weight is uniformly distributed, with a mean of 51.5g and 70% of them less than 52.5g.

What's the probability of mints that weigh less than label weight produced by Machine B?

Would be very grateful to receive any assistance regarding my question. Thank you!

Answer 1

Working in grams, for a $U(a,\,b)$ distribution we have$$a+b=2\mu=103,\,a+0.7(b-a)=0.3a+0.7b=52.5$$(or, if you prefer, $3a+7b=525$). The simultaneous equations have solution $a=49,\,b=54$. So the probability of a value $<50$ is $\frac{50-49}{54-49}=\frac15$.

Answer 2

$20\%$ of this uniform distribution falls between $51.5$ and $52.5$. So the support interval of the distribution should have length $5$, and be centered at the mean (because of uniformity).

Answer 3

You can find the probability directly without calculating the boundaries of the uniform distribution as follows (calculations in g):

• $51.5$ correspond to $50\%$ and $52.5$ correspond to $70\%$
• $\Rightarrow 1 = 52.5-51.5$ correspond to $20\%$
• $51.5 -50 = 1.5$ correspond to $30\%$ $$\Rightarrow 50 \mbox{ correspond to } 50\% - 30\% = \boxed{20\%}$$