Let the sequence Xn, n = 1, 2, 3, ..., be a Bernoulli process with parameter P(Xn = 1) = p for all n >= 1. Let be the time when a value of 0 is first observed: U = min{n: Xn = 0} Then, the random variable is:
A) Geometric with parameter p
B) Geometric with parameter 1 - p
C) None of the above
1-p Geometric with parameter
Well, you know what they say about randomness and geometric shapes... they don't mix! So the answer is C) None of the above. Keep searching for the right shape! Maybe it's more of a rhombus or a trapezoid. Who knows?
To determine the distribution of the random variable U, we can consider it as the number of trials (n) until the first success (Xn = 0).
A geometric distribution models the number of trials until the first success, with each trial having a fixed probability of success.
In this case, the probability of success is P(Xn = 0), which is the probability of observing a value of 0. However, the given information only provides the parameter for P(Xn = 1), not P(Xn = 0).
So, we cannot directly conclude that the distribution of U is geometric with parameter p or 1 - p.
Hence, the correct answer is C) None of the above.
To find the distribution of the random variable U, we need to determine its probability mass function.
Let's consider the probability that U = u, where u is a non-negative integer. This means that the first occurrence of a value of 0 happens at time u.
For U to be equal to u, the sequence X1, X2, ..., Xu-1 must all be equal to 1, and the value at time Xu must be 0.
The probability that X1, X2, ..., Xu-1 are all equal to 1 can be calculated as (1 - p)^(u-1), since each individual observation has a probability of p = P(Xn = 1), and we have (u-1) observations of 1.
The probability that Xu is equal to 0 is simply p = P(Xu = 0).
Therefore, the probability that U = u is given by:
P(U = u) = (1 - p)^(u-1) * p
This matches the probability mass function of a geometric distribution with parameter 1 - p.
Hence, the answer is:
B) Geometric with parameter 1 - p