The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function shown below. Find the probability that a bottle of this medicine will have a shelf life of (a) at least 50 days; (b) anywhere from 30 to 80 days. f(x) = 5,000 (x+50)³ , X>0 elsewhere

Given the probability density function f(x)=15f(x)=15 over the interval [3,8][3,8], find the expected value, the mean, the variance and the standard deviation.Expected value:   Mean:   Variance:   Standard Deviation

The repair time (in hours) for a certain machine is a random variable with probability density function (xe S(x) = What is the probability that the repair time is less than 2 hours? b. a. What is the probability that the repair time is between 1.5 and 3 hours? Find the mean repair time. d. C. Find the cumulative distribution function of the repair times.

Given the probability density function f(x)=13f(x)=13 over the interval [4,7][4,7], find the expected value, the mean, the variance and the standard deviation.Expected value:   Mean:   Variance:   Standard Deviation:

Given the probability density function f(x)=14f(x)=14 over the interval [3,7][3,7], find the expected value, the mean, the variance and the standard deviation.Expected value:   Mean:   Variance:   Standard Deviation:

Suppose the weight of pieces of passenger luggage for domestic airline flights follows a normal distribution with u = 26 pounds and o = 6.9 pounds. (a) Calculate the probability that a piece of luggage weighs less than 31.4 pounds. (Assume that the minimum weight for a piece of luggage is 0 pounds.) X (b) Calculate the weight where the probability density function for the weight of passenger luggage is increasing most rapidly. Xlb (c) Use the Empirical Rule to estimate the percentage of bags that weigh more than 12.2 pounds. X % (d) Use the Empirical Rule to estimate the percentage of bags that weigh between 19.1 and 39.8. X % (e) According to the Empirical Rule, about 84% of bags weigh less than X pounds. Enter an exact number.

If W denotes the weight (in pounds) of an individual and t denotes time (in weeks), then dW/dt is the rate of weight gain or loss (in lb/wk). Suppose that a person weighed 220 pounds 10 weeks ago, and now weighs 198 pounds. Using the Mean Value Theorem, show that their rate of rate loss must have exceeded 2 lb/wk at some point during this 10 week period.

A lamp has two light bulbs with an average lifespan of 1400 hours. Assuming we can model the probability of failure of these bulbs by an exponential density function with mean = 1400, find the probability that both of the lamp's bulbs fail within 800 hours.

For a particular kind of battery, the probability density function for x hours to be the life of a battery selected at random is given by -x/60 x > 0 f(x) = - 60 0, x < 0 Find the probability that the life of a battery selected at random will be between 15 and 25 hours.

An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the chemical is first determinedto be present. At very low concentrations, however, the chemical may fail tobe detected even if it is present. This phenomenon is modeled by assuming thatif the concentration is x, the chemical is detected with probability R(x). Let Ydenote the concentration of a chemical in a soil in which it has been determinedto be present. Show that the density function of Y isg(y) = R(y) f (y)/An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the…