91Ó°ÊÓ

In a large factory, there is an average of two accidents per day, and the time between accidents has an exponential density function with an expected value of \(\frac{1}{2}\) day. Find the probability that the time between two accidents will be more than \(\frac{1}{2}\) day and less than 1 day.

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

Find the value of \(k\) that makes the given function a probability density function on the specified interval. $$f(x)=k, 5 \leq x \leq 20$$

Let \(X\) be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for \(X\) is \(f(x)=20 x^{3}(1-x), 0 \leq x \leq 1.\) (a) Find \(E(X)\) and give an interpretation of this quantity. (b) Compute \(\operatorname{Var}(X).\)

Consider a group of patients who have been treated for an acute disease such as cancer, and let \(X\) be the number of years a person lives after receiving the treatment (the survival time). Under suitable conditions, the density function for \(X\) will be \(f(x)=k e^{-k x}\) for some constant \(k.\) (a) The survival function \(S(x)\) is the probability that a person chosen at random from the group of patients survives until at least time \(x .\) Explain why \(S(x)=1-F(x),\) where \(F(x)\) is the cumulative distribution function for \(X,\) and compute \(S(x).\) (b) Suppose that the probability is .90 that a patient will survive at least 5 years \([S(5)=.90] .\) Find the constant \(k\) in the exponential density function \(f(x).\)

Suppose that the lifetime \(X\) (in hours) of a certain type of flashlight battery is a random variable on the interval \(30 \leq x \leq 50\) with density function \(f(x)=\frac{1}{20}\) \(30 \leq x \leq 50 .\) Find the probability that a battery selected at random will last at least 35 hours.

The time (in minutes) required to complete a certain subassembly is a random variable \(X\) with the density function \(f(x)=\frac{1}{21} x^{2}, 1 \leq x \leq 4.\) (a) Use \(f(x)\) to compute \(\operatorname{Pr}(2 \leq X \leq 3).\) (b) Find the corresponding cumulative distribution function \(F(x).\) (c) Use \(F(x)\) to compute \(\operatorname{Pr}(2 \leq X \leq 3).\)

Time of Birth The gestation period (length of pregnancy) of a certain species is approximately normally distributed with a mean of 6 months and a standard deviation of \(\frac{1}{2}\) month. (a) Find the percentage of births that occur after a gestation period of between 6 and 7 months. (b) Find the percentage of births that occur after a gestation period of between 5 and 6 months.

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.

Let \(X\) be a continuous random variable with the density function \(f(x)=2(x+1)^{-3}, x \geq 0.\) (a) Verify that \(f(x)\) is a probability density function for \(x \geq 0.\) (b) Find the cumulative distribution function for \(X.\) (c) Compute \(\operatorname{Pr}(1 \leq X \leq 2)\) and \(\operatorname{Pr}(3 \leq X).\)