Exam 10: Probability and Calculus

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​ ​

Suppose X is a normal random variable with and . Find . Round your answer to four decimal places. ​

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

Suppose the time intervals in seconds between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed with associated probability density function . ​ Find the probability that the average time interval between arrivals of successive cars is more than 3 sec. Give the answer to four decimal places, if necessary.

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

To be eligible for further consideration, applicants for certain Civil Service positions must first pass a written qualifying examination on which a score of 80 or more must be obtained. In a recent examination, it was found that the scores were normally distributed with a mean of 70 points and a standard deviation of 10 points. Determine the percentage of applicants who passed the written qualifying examination. ​ Round your answer to the nearest hundredth, if necessary. ​ __________%

The quantity demanded x (in thousands of pounds) of a certain brand of butter per week is a continuous random variable with probability density function ​ ​ What is the expected demand for this brand of butter per week? ​ __________ lb/wk

The life span of a certain plant species (in days) is described by the probability density function ​ ​ If a plant of this species is selected at random, how long can the plant be expected to live? ​ __________ days

According to the data released by a city's Chamber of Commerce, the weekly wages of factory workers are normally distributed with a mean of $380 and a standard deviation of $50. Round your answers to four decimal places, if necessary. ​ Find the probability that a worker selected at random from the city has a weekly wage of less than $300. ​ __________ ​ Find the probability that a worker selected at random from the city has a weekly wage of more than $460. ​ __________ ​ Find the probability that a worker selected at random from the city has a weekly wage of between $350 and $450. ​ __________

Suppose the time intervals in seconds between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed with associated probability density function . Find the probability that the average time interval between arrivals of successive cars is more than 9 sec. Round your answer to four decimal places. ​

Find the value of the constant ​k so that the function is a probability density function in the indicated interval. ​

Find the value of the constant k so that the function is a joint probability density function on D. ​

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

According to data released by a city's Chamber of Commerce, the weekly wages (in dollars) of female factory workers are normally distributed with a mean of 470 and a standard deviation of 40. Find the probability that a female factory worker selected at random from the city has a weekly wage of $450 to $550. Round your answer to four decimal places. ​

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​ ​ = __________

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​ ​ = __________

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​ ​

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​ ​ __________

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m. ​ ​

Find the value of the constant k so that the function is a joint probability density function on D. ​