91Ó°ÊÓ

Verify that each of the following functions is a probability density function. \(f(x)=\frac{1}{18} x, 0 \leq x \leq 6\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=\frac{3}{2} x-\frac{3}{4} x^{2}, 0 \leq x \leq 2\)

Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). \(f(x)=\frac{3 \sqrt{x}}{16}, 0 \leq x \leq 4\)

During a certain part of the day, the time between arrivals of automobiles at the tollgate on a turnpike is an exponential random variable with expected value 20 seconds. Find the probability that the time between successive arrivals is more than 60 seconds.

Find the value of \(k\) that makes the given function a probability density function on the specified interval. \(f(x)=k x^{2}(1-x), 0 \leq x \leq 1\)

Probability of an Infection Suppose that a large number of persons become infected by a particular strain of staphylococcus that is present in food served by a fast-food restaurant and that the germ usually produces a certain symptom in \(5 \%\) of the persons infected. What is the probability that, when customers are examined, the first person to have the symptom is the fifth customer examined?

Maximum Likelihood Exercises 19 and 20 illustrate a technique from statistics (called the method of maximum likelihood) that estimates a parameter for a probability distribution. In a production process, a box of fuses is examined and found to contain two defective fuses. Suppose that the probability of having two defective fuses in a box selected at random is \(\left(\lambda^{2} / 2\right) e^{-\lambda}\) for some \(\lambda\). Take first and second derivatives to determine the value of \(\lambda\) for which the probability has its maximum value.

The number of babies born each day in a certain hospital is Poisson distributed with \(\lambda=6.9\). (a) During a particular day, are 7 babies more likely to be born than 6 babies? (b) What is the probability that at most 15 babies will be born during a particular day?

Amount of Milk in a Container If the amount of milk in a gallon container is a normal random variable, with \(\mu=\) \(128.2\) ounces and \(\sigma=.2\) ounce, find the probability that a random container of milk contains less than 128 ounces.

SAT Scores Distribution The Math SAT scores of a recent freshman class at a university were normally distributed, with \(\mu=535\) and \(\sigma=100\). (a) What percentage of the scores were between 500 and \(600 ?\) (b) Find the minimum score needed to be in the top \(10 \%\) of the class.