Question

Improper Integrals. Say that f is integrable on [a,[infinity]) if limb→[infinity]

R b a f(x)dx =: R [infinity] a f(x)dx exists. (a) For which real values of p
does R [infinity] e (logx)^p/ x dx exist? (b) Show that R [infinity] 0 sin

Answer 1

For which real values of p the integral R [infinity] e (logx)^p / x dx exists depends on the behavior of the integrand as x approaches infinity. Continuous probability distributions have a property where P(x = a) is always 0, and the total area under a probability density function equals 1.

Step-by-step explanation:

Evaluating Improper Integrals with Parameters

To determine for which real values of p the improper integral R [infinity] e (logx)^p / x dx exists, we must consider the behavior of the function as x approaches infinity. In the case of the integral from e to infinity of (logx)p / x, analysis involves the comparison test or limit comparison test with known convergent or divergent integrals.

In evaluating a continuous probability distribution function, the probability P(x > 15) for x greater than 15 is 0 if the distribution is defined only for x between 0 and 15. Similarly, P(x = 7) in a continuous distribution is also 0, reflecting the concept that the probability of any single point is zero in a continuous distribution. The area under a probability density function represents the total probability and must equal 1, so the probability P(x < 0) would be 0 if the distribution is defined for non-negative values.