Please answer all 9 questions.

A random variable has CDF given by F, = A i =0, 1,2 1 1 = 3 if A = 0.72, then what is p3 ? Round your answer to two decimal places. Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you win nothing. If you get a spade, you win $94.5. For any club, you win $37.26 plus an extra $39.44 for the ace of clubs. What is the expected value of your winnings if you play the game? Round your answer to five decimal places. In a new card game, you start by paying $61.75. From a well-shuffled full deck of cards draw 3 cards without replacement. If you draw 3 hearts, you win $145.13. If you draw 3 black cards, you win $256.93. For any other draws, you win nothing. What is your expected winnings in this game? Round your answer to five decimal places. Large black-tailed gulls usually lay one to three eggs, and rarely have a fourth egg clutch. It is thought that clutch sizes are limited by how effectively parents can incubate their eggs. Suppose that on average, gulls have a 32.2% of laying 1 egg, 11.5% of laying 2 eggs, 7.7% chance of laying 3 eggs, and 48.6% chance of laying 4 eggs. What is the expected value of the number of eggs laid by a random sample of 100 gulls? Round your answer to five decimal places. A probability density function for the random variable X is given by p, = k()', where k is a constant. What value must k be if X takes on integer values between 1 and n? Hint: )|_, P = 1-2) if 1-,if0 Sx51 O a. ( 4 ) Ob. k = - 1-( 4 ) Ock = 1-(4) Od. k = 1-() Oe. k =- The number of different mammal species observed along a transect through a forest is a random variable X with CDF 0.01, 1 =0 0.16, i = 2 0.36, i = 8 F= 0.71, i = 13 0.96, i = 17 i = 20 What is the expected value of the random variable X? Round your answer to five decimal places. A random variable has CDF given by i = 0, 1,2 F1= 1 1=3 if A = 0.56, then what is p2? Consider the two functions y1 = x - 3 and y2 = b on the interval [-a, a], if a = 3.8. What value does b have to be in order for the area between y, and y2 and [-a, a], to equal 45.1? Round your answer to five decimal places. Consider two functions y1 = f(x) and y2 = g(x), and that y1 / 2 except where they intersect. If these two functions intersect at 275 places, how many different integrals do you need to find the total area enclosed by the two curves