Independent dice throw

Question

I need help on this question regarding dice and probability.

The question is, that a dice throws three times independently of each other. The random variable $X$ denotes the number of sides on the $i$-th throw, where $i$ is $i=1,2,3$. The maximum of the random variable $Y$ is given as $Y_{max}(X_i)$.

I need to find the cdf and pmf of the random variable $Y$.

I don't know how to go on with this question.

Thanks in advance.

Answer 1

Hints:

$P\left(Y\leq k\right)=P\left(X_{1}\leq k\wedge X_{2}\leq k\wedge X_{3}\leq k\right)=P\left(X_{1}\leq k\right)P\left(X_{2}\leq k\right)P\left(X_{3}\leq k\right)$

$P\left(Y=k\right)=P\left(Y\leq k\right)-P\left(Y\leq k-1\right)$

Answer 2

Ok, I think I understand. You need to know the maximum value of a throw with n=3 dice.

I did some time ago a web application with a similar thing where you can see it, formulas and setup: type of dice and number of them, and search for some specific probability, etc.

You can see that, by example, for

$$P(V{=}X)=\frac{1}{D^n}\sum_{k=1}^{n}\binom{n}{k}(X-1)^{n-k}=\frac{X^n-(X-1)^n}{D^n}$$

In this formula V is the maximum value of n throws.

The binomial coefficient represent the amount of permutations between n positions of dice with a value of V and dice with value less than V on the throw. The power $(X-1)^{n-k}$ represent the variations with repetition over $X-1$ possible values on $n-k$ positions.

Finally $\frac{1}{D^n}$ represent total possible throws (variations with repetition over n positions that can take D different values), so dividing cases where V is the maximum value of the throw between all possible cases you have it probability.

From here summing over n, D or any other parameter let to you some distribution or accumulated distribution (anti-derivative).