For each of the following statements, state whether it is true (meaning, always true) or false (meaning, not always true):
1. Let X and Y be two binomial random variables.
(a) If X and Y are independent, then X+Y is also a binomial random variable.
(b) If X and Y have the same parameters, n and p , then X+Y is a binomial random variable.
(c) If X and Y have the same parameter p , and are independent, then X+Y is a binomial random variable.
2. Suppose that, E[X]=0 . Then, X=0 .
3. Suppose that, E[X2]=0 . Then, P(X=0)=1 .
4. Let X be a random variable. Then, E[X2]≥E[X] .
5. Suppose that, X is a random variable, taking positive integer values, which satisfies E[(X−6)2]=0 . Then, pX(4)=pX(5) .
6. Suppose that E[X]≥0 . Then, X≥0 with probability 1, i.e., P(X≥0)=1 .
Answers:
1.
a) False
b) False
c) True
2. False
3. True
4. False
5. True
6. False
answers are all correct
1. a) False - If X and Y are independent binomial random variables, their sum X+Y is not necessarily a binomial random variable. It may have a different distribution.
b) False - If X and Y have the same parameters, n and p, their sum X+Y is not necessarily a binomial random variable. It may have a different distribution.
c) True - If X and Y have the same parameter p and are independent, their sum X+Y is a binomial random variable.
2. False - If the expected value of X is 0, it does not necessarily mean that X is always equal to 0. X can have other values as well.
3. True - If the expected value of X^2 is 0, it implies that X^2 is always equal to 0. Therefore, the probability that X is equal to 0 is 1.
4. False - The expected value of X^2 does not necessarily have to be greater than or equal to the expected value of X. It depends on the distribution of X.
5. True - If the expected value of (X-6)^2 is 0, it implies that (X-6)^2 is always equal to 0. Therefore, the probabilities of X being 4 and 5 are the same.
6. False - If the expected value of X is greater than or equal to 0, it does not necessarily mean that X is always greater than or equal to 0. X can have negative values as well.
1.
a) False - If X and Y are independent, their sum X+Y generally follows a binomial distribution only if X and Y have the same parameters (n and p).
b) False - X+Y is not necessarily a binomial random variable even if X and Y have the same parameters (n and p). It depends on the specific relationship between X and Y.
c) True - If X and Y have the same parameter p and are independent, their sum X+Y follows a binomial distribution.
2. False - The expected value (E[X]) being 0 does not imply that X will always be 0. It means that, on average, X is expected to be 0, but it can take other values as well.
3. True - If the expected value of X squared (E[X^2]) is 0, it implies that X is almost surely 0. In other words, the probability of X being 0 (P(X=0)) is 1.
4. False - The expected value of X squared (E[X^2]) does not necessarily have to be greater than or equal to the expected value of X (E[X]). The two values depend on the specific distribution and can be different.
5. True - If E[(X-6)^2] = 0, it means that X-6 is almost surely 0. This implies that p(X=4) = p(X=5) = 1, since the only possible values of X are 4 and 5.
6. False - The expected value of X (E[X]) being greater than or equal to 0 does not guarantee that X is greater than or equal to 0 with probability 1 (P(X >= 0) = 1). It means that, on average, X is expected to be greater than or equal to 0, but there can still be some probability for X to be negative.
To determine whether each of the statements is true or false, we need to understand the concepts and properties involved. Let's go through each statement and explain how to determine its truth value.
1.
(a) The statement says that if X and Y are independent binomial random variables, then the sum of X and Y, denoted by X+Y, is also a binomial random variable. To determine its truth value, we need to know that the sum of independent binomial random variables is not always binomial. Therefore, the statement is false.
(b) The statement claims that if X and Y have the same parameters (n and p), then their sum X+Y is a binomial random variable. To check its truth value, we need to confirm that the sum of binomial random variables with the same parameters is always binomial. Since this is true, the statement is false.
(c) The statement states that if X and Y have the same parameter p, and they are independent, then X+Y is a binomial random variable. To evaluate its truth value, we must verify that the sum of independent random variables with the same parameter p is always binomial. This property holds, so the statement is true.
2. The statement asserts that if the expected value of X is 0, then X must be equal to 0. To determine its truth value, we need to understand that the expected value being 0 only implies that X is centered around 0, but it does not guarantee that X is always 0. Therefore, the statement is false.
3. The statement claims that if the expected value of X^2 is 0, then the probability of X being 0 is 1. To check its truth value, we need to know that the expected value being 0 implies that X^2 is always 0, which means X must be 0 with probability 1. Therefore, the statement is true.
4. The statement states that the expected value of X^2 is greater than or equal to the expected value of X. To determine its truth value, we need to understand that the expected value of X^2 measures the spread or variability of X, while the expected value of X measures its average value. Generally, the spread of X^2 is greater than or equal to the spread of X, so this statement is false.
5. The statement asserts that if X is a random variable taking positive integer values such that the expected value of (X-6)^2 is 0, then the probability of X being 4 is equal to the probability of X being 5. To evaluate its truth value, we need to know that (X-6)^2=0 implies that X must be either 4 or 5. Therefore, the statement is true.
6. The statement claims that if the expected value of X is greater than or equal to 0, then X must be greater than or equal to 0 with probability 1, i.e., P(X≥0)=1. To determine its truth value, we need to understand that the expected value being greater than or equal to 0 only implies that X is non-negative on average, but it does not guarantee that X is always non-negative. Therefore, the statement is false.