Determine whether the distribution is a probability distributi...

Question

Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

Table displaying values of x and their corresponding probabilities p(x) for discrete random variables.

Accepted Answer

Welcome back, everyone. Refer to the following distribution. Is it a valid probability distribution? If not, explain your reasoning. We're given for answer choices. A says, yes, it is a probability distribution because the probabilities add up to 1. No, it is not a probability distribution because all probabilities must be greater than 0.5. No, it is not a probability distribution because One probability is negative, and yes, it is a probability distribution because all probabilities are valid. So let's break all two criteria for a probability distribution. First all, each individual probability must be between 0 and 1 inclusive. So if we look at our table, we have 0.2, 0.5, 0.4, and negative 0.1. So the first three ones, they fall within the range, but the final one is negative, right? So we can immediately say that this cannot be probability distribution because the first criterion is not met. We have a negative value, right? Now, the second criterion is to ensure that the sum of all probabilities is equal to 1. So what we're going to do is simply identify the sum of our probabilities, that would be probability 1 + 2 + 3. Plus 4 we have 4 of them, right? The first probability for the number of bats of zero is 0.2. The second one is 0.5, followed by 0.4, and then we're adding -0.1. This would be equal to 1, right? So the second criterion is math, however, We really want to ensure that both of them are met. So as long as one of them is not met, we can immediately conclude that this would not be a valid probability distribution. So now considering the answer choices, A says yes, it is a probability distribution because the probabilities add up to one. While this fact is correct. The probability distribution also needs its probabilities, specifically individual probabilities, to fall within the range between 0 and 1 inclusive. So it's not sufficient to, it's not sufficient to have a sum of probabilities equal to 1. We can also exclude the it says yes, because all probabilities are valid. We know that. The final probability for the number of bats of 3 is invalid. It's negative, and now between B and C. C says no, it is not a probability distribution because one probability is negative. This is exactly what we need and B no, it is not a probability distribution because all probabilities must be greater than 0.5. We don't have such a requirement for probability distributions. So the correct answer to this problem corresponds to the answer to C. Thank you for watching.

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