If the moment-generating function of X is given by

$$M(t) = e^{500t + 5000t^2}$$

, find P[27, 060 ≤ (X - 500)² ≤ 50,240].

If the moment-generating function of X is

$$M(t) = 2/5 e^t + 1/5 e^{2t} + 2/5 e^{3t}$$

, find the mean, variance, and pmf of X.

If the moment-generating function of a random variable W is

$$M(t) = (1 - 7t)^{-20}$$

, find the pdf, mean, and variance of W.

If X is normally distributed with a mean of 6 and a variance of 25, find a. P(6 ≤ X ≤ 12). b. P(0 ≤ X ≤ 8). c. P(−2 < X ≤ 0). d. P(X > 21). e. P(|X − 6| < 5). f. P(|X − 6| < 10) g. P(|X − 6| < 15). h. P(|X − 6| < 12.41).

Cars arrive at a tollbooth at a mean rate of 5 cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.30 minutes before collecting the eighth toll.

What are the pdf, the mean, and the variance of X if the moment-generating function of X is given by the following? a. M(t) = 1/1 - 3t, t < 1/3. b. M(t) = 3/3 - t, t < 3.

The pdf of Y is g(y) = d/y³, 1 < y < ∞. a. Calculate the value of d so that g(y) is a pdf. b. Find E(Y). c. Show that Var(Y) is not finite.

The serum zinc level X in micrograms per deciliter for males between ages 15 and 17 has a distribution that is approximately normal with μ = 90 and σ = 15. Compute the conditional probability P(X > 120 | X > 105).

The moment generating function of X is given by

$$M _ { X } ( t ) = \exp \left\{ 2 e ^ { t } - 2 \right\}$$

and that of Y by

$$M _ { Y } ( t ) = \left( \frac { 3 } { 4 } e ^ { t } + \frac { 1 } { 4 } \right) ^ { 10 }$$

. If X and Y are independent, what are (a) P{X+Y=2}? (b) P{XY=0}? (c) E[XY]?

6) If the moment-generating function of a random variable $$ x $$ is $$ \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } e ^ { 2 } \right) ^ { 5 } $$ , find $$ P _ { T } ( X = 2 \text { or } 3 $$ ). 7) The moment-generating function of a random variable $$ x $$ is $$ \left( 3 + \frac { 1 } { 3 } e ^ { 2 } $$ . Show that \right. $$ \left. \qquad P x ( \mu - 2 a < X < \mu + 2 \alpha ) = \sum ^ { 5 } \text { ( } ^ { 9 } \right) \left( \frac { 1 } { 2 } \right) ^ { 2 } ( 2 ) ^ { 0 - 2 } $$

Use integration to find a general solution of the differential equation.

$$dy/dx = e^2-^x)$$

$$\text{ The moment-generating function of a certain Poisson random variable } X \text{ is given by }$$

$$M_X(t) = e^{4(e^t−1)}.$$

$$\text{Find } P(\mu − 2\sigma < X < \mu + 2\sigma).$$

find the number of distinguishable permutations of the group of letters. B, A, S, K, E, T, B, A, L, L

Let the distribution of X be N(μ, σ²). Show that the points of inflection of the graph of the pdf of X occur at x = μ ± σ.