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A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function $$f(x)=\left\\{\begin{array}{ll}5(1-x)^{4} & 0

The lifetime in hours of an electronic tube is a random variable having a probability density function given by $$f(x)=x e^{-x} \quad x \geq 0$$ Compute the expected lifetime of such a tube.

A point is chosen at random on a line segment of length \(L\). Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than \(\frac{1}{4}\)

You arrive at a bus stop at 10 o'clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10: 30 . (a) What is the probability that you will have to wait longer than 10 minutes? (b) If, at \(10: 15,\) the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

Let \(X\) be a normal random variable with mean 12 and variance 4. Find the value of \(c\) such that \(P\\{X>c\\}=.10\)

Suppose that the height, in inches, of a 25 -year-old man is a normal random variable with parameters \(\mu=71\) and \(\sigma^{2}=6.25 .\) What percentage of 25 year-old men are over 6 feet, 2 inches tall? What percentage of men in the 6 -footer club are over 6 feet, 5 inches?

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability.95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Twelve percent of the population is left handed. Approximate the probability that there are at least 20 left-handers in a school of 200 students. State your assumptions.

A model for the movement of a stock supposes that if the present price of the stock is \(s\), then, after one period, it will be either \(u s\) with probability \(p\) or \(d s\) with probability \(1-p .\) Assuming that successive movements are independent, approximate the probability that the stock's price will be up at least 30 percent after the next 1000 periods if \(u=1.012, d=0.990,\) and \(p=.52\)

The lung cancer hazard rate \(\lambda(t)\) of a \(t\) -year-old male smoker is such that $$\lambda(t)=.027+.00025(t-40)^{2} \quad t \geq 40$$ Assuming that a 40 -year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50 and (b) age 60 without contracting lung cancer?