A catalyst researcher states that the diameters, in microns, of the pores in a new product she has made have the exponential distribution with parameter λ = 0.25. Suppose that the diameters of 10 pores are independent. a) Find the probability that more than seven of them have diameters less than 3 microns. b) Find the probability that exactly one of them has a diameter greater than 11 microns.

A person arrives at the side of a single lane, one way street as a car passes by. Consider this time as time 0. There is no traffic lights and no pedestrian lanes in the nearby area. He stands there watching cars pass by. The number of cars to pass by within a period of time cane be modeled using a poison distribution. Suppose the cars pass by the person at a rate of 4.5 cars per minute. Assume that the cars pass by instantaneously. what is the probability that no cars pass by him within a 2 minute window b) if the person had to guess how many cars would pass by in the next five minutes, what number should he guess? Why? c) what is the distribution for the time until the first car passes by him(time is continuous). What is the expected time until the first car passes by him?

2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs 2 S (v) . else What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density а. function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: Find?(r - 1.5)) = 1 CO0

In a trial to compare a stannous fluori de dentifrice (A) with a commerciall y available fluoride-free dentifrice (D), 270 children received A and 250 recei ved D for a period of 3 years. The nu mber x of DMFS increments (i.e., the number of new decayed, missing, an d filled tooth surfaces) was obtaine d for each child. Do the results provid e strong enough evidence to suggest a real effect of fluoride in reducing e mean DMFS?Results were: Dentifrice A: ₁ = 9.78 SA = 7.51. Dentifrice D: x = 12.83 SD = 8.31.