Let X and Y be real-valued random variables, both distributed according to a distribution \, \mathbf{P}.\, (We make no assumption about their joint distribution). Let F denote the cdf of \, \mathbf{P}\,.
Which of the following are true about the cdf F? (Choose all that apply.)
P(X \leq t) and P(Y \leq t) are random variables.
For all t \in \mathbb {R}, F(t) = P(X \leq t) and F(t) = P(Y \leq t).
F(t) = P(X \leq t) = P(Y \leq t) only if X and Y are independent.
\displaystyle \lim _{t \to \infty } F(t) = 1.
\displaystyle \lim _{t \to -\infty } F(t) = 0.
\displaystyle \int _{-\infty }^{\infty } F(t) dt =1
unanswered
The correct options are:
1. P(X <= t) and P(Y <= t) are random variables.
4. lim (t -> infinity) F(t) = 1.
5. lim (t -> -infinity) F(t) = 0.
6. ∫(-infinity to infinity) F(t) dt = 1.
Option 2 is incorrect because F(t) = P(X <= t) and F(t) = P(Y <= t) are both true, but they don't have to be equal.
Option 3 is incorrect because F(t) = P(X <= t) = P(Y <= t) does not necessarily imply that X and Y are independent.
For the given statements about the cdf F:
1. P(X ≤ t) and P(Y ≤ t) are random variables: This statement is false. P(X ≤ t) and P(Y ≤ t) are probabilities and not random variables.
2. For all t ∈ ℝ, F(t) = P(X ≤ t) and F(t) = P(Y ≤ t): This statement is true. The cdf F is defined as F(t) = P(X ≤ t) and also as F(t) = P(Y ≤ t) for any value of t.
3. F(t) = P(X ≤ t) = P(Y ≤ t) only if X and Y are independent: This statement is true. If X and Y are independent random variables, then the joint distribution factors into the product of their marginal distributions, and as a result, F(t) = P(X ≤ t) = P(Y ≤ t). However, this equation may not hold if X and Y are dependent.
4. lim t→∞ F(t) = 1: This statement is true. According to the properties of a cdf, as t approaches infinity, the probability of X or Y being less than or equal to t approaches 1. Hence, the limit of the cdf F as t approaches infinity is 1.
5. lim t→-∞ F(t) = 0: This statement is true. Similarly, as t approaches negative infinity, the probability of X or Y being less than or equal to t approaches 0. Therefore, the limit of the cdf F as t approaches negative infinity is 0.
6. ∫-∞∞ F(t) dt = 1: This statement is false. The integral of the cdf F over the entire real line is not necessarily equal to 1. The integral of a cdf represents the area under the curve, which can be less than or equal to 1 depending on the distribution.