Calculating the PMF of Min. Observation of Discrete Distribution Sample Size 5

[happyg1]

Hi,
I know this isn't hard, but I have a block on it.
Here's the question:
Let $$f(x)=1/6, x=1,2,3,4,5,6$$ zero elsewhere be the pmf of a distribution of the discrete type. Show that the pmf of the smallest observation of a random sample of size 5 from this distribution is:
$$g_1(y_1)=\left(\frac{7-y_1}{6}\right)^5-\left(\frac{6-y_1}{6}\right)^5,y_1=1,2,3,4,5,6$$

I know I'm looking for the pmf of the minimum, which is the y_1, and I did this:
$$P(Y_1=y_1)=P(Y_1\leq y_1)-P(Y_1<y_1)$$
I'm having trouble seeing why there's a 7 in the solution. I'm missing something.
I know that wheneve I get the probabilities of each piece, they will be to the 5th power because there's 5 observations in the sample and we multiply the probabilities together. What the probabilities are is the part that has me confused.
We've been focusing on continuous problems for so long I can't wrap my mind around this.
Any hints or pointers will be appreciated.

 

[kurt.physics]

Thanks!The 7 in the solution is coming from the number of possible outcomes for the minimum observation. Since you have 6 possible values for the observed variable (1,2,3,4,5,6), there are 7 possibilities for the minimum observation: 1,2,3,4,5,6, and 0. The probability that the minimum observation is 0 is 0, so it can be excluded from the equation. So the equation is P(Y_1=y_1) = P(Y_1≤y_1) - P(Y_1<y_1). For the first part, P(Y_1≤y_1), this is the probability that the minimum observation is less than or equal to a certain value (y_1). To calculate this, you need to take the probability for each of the possible values (1,2,3,4,5,6) and multiply them together. This gives you the probability that all of the observations are less than or equal to y_1. For the second part, P(Y_1<y_1), this is the probability that the minimum observation is less than a certain value (y_1). To calculate this, you need to take the probability for each of the possible values (1,2,3,4,5,6) and multiply them together, except for the probability of y_1. This gives you the probability that all of the observations are less than y_1. Combining these two equations gives you the solution above.