Final answer:
To determine the constant c, we find the sum of all probabilities for all possible values of k. This leads to c = 1/2. P(X > 2) = 0, and F(t) = P(X <= t) can be computed by summing the probabilities for all values of k from 0 to t.
Step-by-step explanation:
To determine the constant c, we need to find the sum of all probabilities for all possible values of k, which should add up to 1. So:
P(X = 0) + P(X = 1) + P(X = 2) + ... = c/0! + c/1! + c/2! + ... = 1.
Since 0! = 1, we can rewrite the equation as:
c + c/1 + c/2 + ... = 1.
This is an infinite geometric series. The sum of an infinite geometric series with a common ratio less than 1 is given by the formula:
Sum = a/(1 - r), where a is the first term and r is the common ratio.
In this case, a = c and r = 1/2. Substituting these values into the formula:
Sum = c/(1 - 1/2) = c/(1/2) = 2c.
Since the sum of all probabilities is equal to 1, we have 2c = 1, which implies c = 1/2.
To compute P(X > 2), we need to find the probability that X takes on a value greater than 2. We can compute this by subtracting the sum of probabilities for X = 0, X = 1, and X = 2 from 1. So:
P(X > 2) = 1 - P(X <= 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)).
Substituting the values of c and the probabilities from the given law:
P(X > 2) = 1 - (1/2 + 1/2 + 1/(2*2)) = 1 - (1/2 + 1/2 + 1/4) = 1 - 1 = 0.
To find F(t) = P(X <= t), we need to find the probability that X takes on a value less than or equal to t. We can compute this by summing the probabilities for all values of k from 0 to t. So:
F(t) = P(X <= t) = P(X = 0) + P(X = 1) + ... + P(X = t).
Substituting the values of c and the probabilities from the given law:
F(t) = 1/2 + 1/2t + 1/(2t(t-1)) + ... + 1/(2t!)