Questions from Calculus

Q: Let a and r be given nonzero numbers. (a

Let a and r be given nonzero numbers. (a) Show that (1 - r)(a + ar + ar2 + … + arn) = a - arn+1, and from this conclude that, for r  1, a + ar + ar2 + … + arn = a/(1 – r) - arn+1/(1 – r). (b) Use the...

Q: For any positive number k, verify that f (x)

For any positive number k, verify that f (x) = 1/(2k3) x2 e-x/k, x ≥ 0, is a density function.

Q: A point is selected at random from the rectangle of Fig.

A point is selected at random from the rectangle of Fig. 2; call its coordinates (u, y). Find the probability that y ≤ sin θ. Figure 2:

Q: Let X be a continuous random variable with density function f (

Let X be a continuous random variable with density function f (x) = 3x-4, x ≥ 1. Compute E(X ) and Var(X ).

Q: If X is a random variable with density function f (x

If X is a random variable with density function f (x) on A ≤ x ≤ B, the median of X is that number M such that ∫AMf (x) dx = 1/2. In other words, Pr (X ≤ M ) = 1/2. Find the median of the random varia...

Q: If X is a random variable with density function f (x

If X is a random variable with density function f (x) on A ≤ x ≤ B, the median of X is that number M such that ∫AMf (x) dx = 1/2. In other words, Pr (X ≤ M ) = 1/2. Find the median of the random varia...

Q: The machine component described in Exercise 11 has a 50% chance

The machine component described in Exercise 11 has a 50% chance of lasting at least how long? Exercise 11: The useful life (in hundreds of hours) of a certain machine component is a random variable X...

Q: In Exercise 12, find the length of time T such that

In Exercise 12, find the length of time T such that half of the assemblies are completed in T minutes or less. Exercise 12: The time (in minutes) required to complete an assembly on a production line...

Q: In Exercise 20 of Section 12.2, find the length

In Exercise 20 of Section 12.2, find the length of time T such that about half of the time you wait only T minutes or less in the express lane at the supermarket. Exercise 20: At a certain supermarke...

Q: Find the number M such that, half of the time,

Find the number M such that, half of the time, the dairy in Exercise 16 sells M thousand gallons of milk or less. Exercise 16: Find Pr (1.5 ≤ X ≤ 1.7) when X is a random variable whose density functi...