Discrete Probability Problem: determining probability mass function and cumulative distribution function Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar.

a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3, and the two individuals #4 and #5.]

b) Obtain the cumulative distribution function of X, and use it to calculate P(2 &lt;= X &lt;= 6).
Let $L_i$ be an indicator function, equal to 1 if #i is late. Then 
$$
    X = 2 L_1 + 2 L_2 + 2 L_3 + L_4 + L_5
$$
where each $L_i$ is independent and follows Bernoulli distribution with probability $p=0.4$. The easiest way to find probability mass function is through the probability generating function:
$$
  \mathcal{P}_X(s) = \mathbb{E}( s^X ) = \left(\mathcal{P}_L(s^2)\right)^3 \cdot \left(\mathcal{P}_L(s)\right)^2 = (1-p+p s^2)^3 (1-p + p s)^2
$$ 
Now expand this polynomial. Then
$$
    \mathbb{P}(X= k) = [s]^k \mathcal{P}_X(s)
$$
If you can use excel simply use the binomial distribution function: BINOM.DIST(x,n,p,cumulative).

Where x is the number of late people, n is the total number of people, p is the probability of someone being late, and cumulative is a bool that says whether or not to calculate the cumulative probability. Consider couples to be one person, since they arrive together. 

So for the pmf use BINOM.DIST with cumulative set to false on each of the x values,
then to find the cdf repeat with cumulative set to true.

This will work up to 5 then you have to use some logic to find the rest from the values you already have.