The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given by

$$f(x)=\left\{\begin{array}{ll}{\lambda e^{-x / 100}} & {x \geq 0} \\ {0} & {x<0}\end{array}\right.$$

What is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will function less than 100 hours?

Circle each preposition and underline each prepositional phrase. Grissom's Mercury flight was like Shepard's suborbital flight.

The distribution function of the random variable X is given

$$F(x)=\left\{\begin{array}{ll}{0} & {x<0} \\ {\frac{x}{2}} & {0 \leq x<1} \\ {\frac{2}{3}} & {1 \leq x<2} \\ {\frac{11}{12}} & {2 \leq x<3} \\ {1} & {3 \leq x}\end{array}\right.$$

(a) Plot this distribution function. (b) What is $P\left\{X>\frac{1}{2}\right\} ?$ (c) What is $P\{2<X \leq 4\} ?$ (d) What is $P\{X<3\} ?$ (e) What is $P\{X=1\} ?$

Suppose that the average number of cars abandoned weekly on a certain highway is 2.2. Approximate probability that there will be (a) no abandoned cars in the next week; (b) at least 2 abandoned cars in the next week.

Let $X$ be a continuous random variable with a probability density function.

$$f(x)=2x,\quad 0\leq x\leq 1.$$

Find the expected value of $X$.

How many ways are there to pack nine identical DVDs into three indistinguishable boxes so that each box contains at least two DVDs?

The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by

$$f(x)=\left\{\begin{array}{ll}{0} & {x \leq 100} \\ {\frac{100}{x^{2}}} & {x>100}\end{array}\right.$$

What is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events $E_i , i = 1, 2, 3, 4, 5,$ that the ith such tube will have to be replaced within this time are independent.

Show that if $E \subset F$ then $P(E) \leq P(F)$. (Hint: Write F as the union of two mutually exclusive events, one of them being E.)

Find the characteristic equation, the eigen-values, and bases for the eigen spaces of the matrix. [1 0 -2, 0 0 0, -2 0 4]

If the density function of X equals

$$f(x)=\left\{\begin{array}{ll}{c e^{-2 x}} & {0<x<\infty} \\ {0} & {x<0}\end{array}\right.$$

find c. What is $P\{X>2\} ?$

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter $\frac{1}{20}$. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed, but rather is (in thousands of miles) uniformly distributed over (0, 40).

The strength X of a certain material is such that its distribution is found by

$$X = e^Y$$

, where Y is N(10, 1). Find the cdf and pdf of X, and compute P(10, 000 < X < 20, 000). Note:

$$F(x) = P(X ≤ x) = P(e^Y ≤ x) = P(Y ≤ 1n x)$$

so that the random variable X is said to have a lognormal distribution.

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman. (For instance, X=1 if the top-ranked person is female.) Find P{X = i}, i=1, 2, 3, ..., 8, 9, 10.

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter $\lambda=1$. (a) What is the probability that a repair time exceeds 2 hours? (b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2 hours?