What is the distribution followed by Y1,...,Yn in this scenario, where Yi = 1 if Xi ≤ 0.5?
Hint: The possible answer choices are:
- Bernoulli
- Poisson
- Normal
- Exponential
In terms of θ, what would be the parameter mθ of this distribution?
To determine the distribution followed by Y1,...,Yn, we need to understand the relationship between Yi and Xi as provided in the description. Here, we are given that Yi equals 1 if Xi is less than or equal to 0.5. This implies that Yi follows a Bernoulli distribution.
The Bernoulli distribution is a discrete probability distribution that represents a random variable that takes only two possible outcomes: success (usually denoted as 1) or failure (usually denoted as 0). In this scenario, Yi can be interpreted as a success (1) if the condition Xi ≤ 0.5 is satisfied; otherwise, it is a failure (0).
Now, let's move on to the second part of the question. The parameter mθ of the distribution represents the probability of success (p).
In a Bernoulli distribution, the probability of success (p) is typically denoted as θ. However, to represent the parameter mθ, we need to understand the relationship between m and θ.
The parameter mθ represents the expected value or mean of the distribution. For a Bernoulli distribution, the expected value is equal to the probability of success. Therefore, in this case, mθ corresponds to the probability of Xi ≤ 0.5, which is θ.
In summary:
- The distribution followed by Y1,...,Yn is Bernoulli.
- The parameter mθ of this distribution is simply θ, representing the probability of success.