I have a question about how to solve this exercise: The monthly sales of a certain consumables store are distributed evenly, with average 4000e and standard deviation 1200e, and the estimated expenses for the same period amount to 4200e.
1-Let X be the random variable number of months with losses in the next semester (6 months). Determine its probability function.
2-What is the probability that in the next semester you will have losses in at least three months?
Here, first of all I have calculated a uniform distribution U (a, b). As the mean and typical deviation indicate, I calculate the values that take a and b with the formulas and they give me [a = 6078.460969 b = 1921.539030, a = 1921.539030 b = 6078.460969].
As a < b , I throw away the first option and I'll take the second one. Once arrived at this point, I do not know how to continue solving the exercise. I have tried with a binomial distribution of parameter 6 but its formula is B (n, p), which I think I only have p. Is it well done? How can I continue solving it?
For section 2, once I have the previous distribution, I have to calculate that Pr {X> 2} Is it correct?
I agree that "distributed evenly" means uniformly. Let $S$ be the number of sales in a given month.
$$S \sim \operatorname{Uniform}(a,b)$$
Then we know $E[S] = \dfrac{1}{2}(a+b)$ and $Var(S) = \dfrac{1}{12}(b-a)^2$. Solving for $a,b$ when $E[S] = 4000$ and $Var[S] = 1200^2$ yields $a=1921.539030, b = 6078.46096$ (with one extraneous solution).
In any given month, with fixed cost $4200$, the probability of a loss is $$P(S \leq 4200)$$
Once you have found that, let $X$ be number of losses in six months.
$$X \sim \operatorname{Binomial}(6, P(S \leq 4200))$$
Then you want to find $P(X \geq 3)$. Can you finish it from here?