Exam 10: Probability and Calculus

Find the value of the probability of the standard normal variable Z. 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. Round the answer to two decimal places, if necessary. ​ ​

Let Z be the standard normal variable. Round your answers to the nearest hundredth, if necessary. ​ Find the value of z if z satisfies . ​ __________ ​ Find the value of z if z satisfies . ​ __________

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 . Round your answers to four decimal places, if necessary. ​ Find . ​ __________ ​ Find . ​ __________ ​ Find . ​ __________

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 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. ​ ; ​

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 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. Round the answer to two decimal places, if necessary. ​ ;

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 50. 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 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. ​ ​ ​ ​ ​ __________

National Welding Company uses industrial robots in some of its assembly-line operations. Management has determined that the lengths of time in hours between breakdowns are exponentially distributed with probability density function ​ ​ What is the probability that a robot selected at random will break down between 500 and 800 hr of use? Round the answer to the nearest hundredth. ​ __________ ​ What is the probability that a robot will break down after 1,100 hr of use? Round the answer to the nearest hundredth. ​ __________

The tread lives of Super Titan radial tires under normal driving conditions are normally distributed with a mean of 42,000 mi and a standard deviation of 5,000 mi. What is the probability that a tire selected at random will have a tread life of more than 35,000 mi? Round your answer to four decimal places, if necessary. ​ __________ ​ If four new tires are installed on a car and they experience even wear, determine the probability that all four tires still have useful tread lives after 35,000 mi of driving. Round your answer to four decimal places, if necessary. ​ __________

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. ​ ​

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function ​ ​ Find the amount of snowfall one can expect in any given month of January in Alaska. ​

National Welding Company uses industrial robots in some of its assembly-line operations. Management has determined that the lengths of time in hours between breakdowns are exponentially distributed with probability density function . ​ What is the probability that a robot selected at random will break down between 400 and 500 hr of use? Round your answer to the nearest hundredth. ​