Let X be the number of independent coin tosses it takes to see the first head, where the coin has probability p of landing on its head. That is, X is a geometric random variable with parameter p. a) Use the above characterization to write the probability mass function of X. b) Show that X has the memoryless property: P(X>= k+l | X>=l) = P(X>=k) for every pair of integers k, l >=0. ( This can be done in two ways, one computational, and one via a soft argument.)
Question
Let X be the number of independent coin tosses it takes to see the first head, where the coin has probability p of landing on its head. That is, X is a geometric random variable with parameter p. a) Use the above characterization to write the probability mass function of X. b) Show that X has the memoryless property: P(X>= k+l | X>=l) = P(X>=k) for every pair of integers k, l >=0. ( This can be done in two ways, one computational, and one via a soft argument.)
Solution
a) The probability mass function (pmf) of a geometric random variable X is given by:
P(X = k) = (1-p)^(k-1) * p, for k = 1, 2, 3, ...
This equation represents the probability that the first success (in this case, the coin landing on heads) occurs on the kth trial. The term (1-p)^(k-1) represents the probability of getting k-1 failures (tails) in a row, and p represents the probability of getting a success (head) on the kth trial.
b) The memoryless property of a random variable states that the probability of an event occurring in the future is independent of past events. For a geometric random variable X, this can be written as:
P(X >= k+l | X >= l) = P(X >= k)
To prove this, we can use the definition of conditional probability:
P(X >= k+l | X >= l) = P(X >= k+l and X >= l) / P(X >= l)
Since the event {X >= k+l} is a subset of the event {X >= l}, we have:
P(X >= k+l and X >= l) = P(X >= k+l)
So,
P(X >= k+l | X >= l) = P(X >= k+l) / P(X >= l)
Using the pmf of X, we can write this as:
P(X >= k+l | X >= l) = [(1-p)^(k+l-1)] / [(1-p)^(l-1)]
Simplifying, we get:
P(X >= k+l | X >= l) = (1-p)^(k-1) = P(X >= k)
This proves that the geometric random variable X has the memoryless property.
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