The expected value of a non-negative continuous random variable X, which is defined by
E(X) = ∫ x * fx(x) dx with lower limit 0 and upper limite infinite
also satisfies E(X) = ∫ P(X > t) dt with lower limit 0 and upper limite infinite
True or false?
True.
This is known as the complementary cumulative distribution function (CCDF) or reliability function, and it represents the probability that X is greater than some threshold t. By integrating this function over all possible values of t, we obtain the expected value of X. This relationship is known as the "tail integral formula" or "survival function formula" for the expected value.
True.
The expected value of a non-negative continuous random variable X can be calculated using the formula:
E(X) = ∫[0,∞] x * fX(x) dx
where fX(x) is the probability density function (pdf) of X.
Alternatively, the expected value can also be expressed in terms of the survival function:
E(X) = ∫[0,∞] P(X > t) dt
The survival function, denoted as S(t), represents the probability that X is greater than t.
By integrating the tail probabilities of X, the expected value can be obtained. Both formulas are equivalent and give the same result.