Show that the following function is a density function. Find the corresponding distribution function. f(x) = - 11x 11e for x>0 = for x 0 for x> The corresponding distribution function is F(x) = for x Let X be a continuous random variable with the following density function. Find E(X) and var(X). f(x) = E(X) = { 7x 5e for x>0 0 for x 0 Suppose that a quantitative character is normally distributed with mean = 13.9 and standard deviation = 3.2. Find an interval centered at the mean such that 95% of the population falls into the interval. Do the same for 99% of the population. The interval for 95% of the population is (Round to one decimal place as needed.) Assume that a quantitative character is normally distributed with mean and standard deviation . Determine what fraction of the population falls into the given interval. [, ) About % of the population falls into the interval. (Round to the nearest whole number as needed.) Suppose that X is normally distributed with mean = 2 and standard deviation 1. For parts (a) through (d), use a table of the standard normal distribution to find x such that each statement holds. (a) P(X x)=0.8 (b) P(X>x) 0.3 (c) P(Xsx) 0.3 (d) P(X-2x)= 0.4 Click the icon to view the table of the standard normal distribution. (a) x= (Round to two decimal places as needed.) In a study of a common insect, the number of bristles on the fifth abdominal sternite was shown to follow a normal distribution with mean 17.4 and standard deviation 2.1. (a) What percentage of the male population has fewer than 15 abdominal bristles? (b) Find an interval centered at the mean so that 90% of the male population have bristle numbers that fall into this interval. Click the icon to view the table of the standard normal distribution. (a) Use the transformation z = X-P to find the value of z. z= (Round to two decimal places as needed) Suppose the weight of an adult animal is normally distributed with mean 3655 g and standard deviation 587 g. What percentage of the population has a weight that exceeds 5200 g? Click the icon to view the table of the standard normal distribution. Use the transformation z= X-P to find the value of z. z= (Round to two decimal places as needed.) Suppose that X is normally distributed with mean - 3 and standard deviation 2. Find P(-6.6x -1.9). Click the icon to view the table of the standard normal distribution. P(-6.6x -1.9) = (Round to four decimal places as needed.)