Chapter 2 Verify that the following functions are probability mass functions, and determine the requested probabilities. 1. a) b) c) d) π·(πΏ ≤ π) π·(πΏ > −π) π·(−π ≤ πΏ ≤ π) π·(πΏ ≤ −π ππ πΏ = π) Solution: - All probabilities are greater than or equal to zero and sum to one. a) π(π ≤ 2) = 1/8 + 2/8 + 2/8 + 2/8 + 1/8 = 1 b) π(π > −2) = 2/8 + 2/8 + 2/8 + 1/8 = 7/8 c) π(−1 ≤ π ≤ 1) = 2/8 + 2/8 + 2/8 =6/8 = 3/4 d) π(π ≤ −1 ππ π = 2) = 1/8 + 2/8 +1/8 = 4/8 =1/2 2. π(π) = ππ+π ππ , x = 0,1,2,3,4 a) π·(πΏ = π) b) π·(πΏ ≤ π) c) π·(π ≤ πΏ < π) d) π·(πΏ > −ππ) Solution: - Probabilities are nonnegative and sum to one a) π(π = 4) = 9/25 b) π(π ≤ 1) = 1/25 + 3/25 = 4/25 c) π(2 ≤ π < 4) = 5/25 + 7/25 = 12/25 d) π(π > −10) = 1 3. The thickness of wood panelling (in inches) that a customer orders is a random variable with the following cumulative distribution function: Determine the following probabilities a) π·(πΏ ≤ π/π) b) π·(πΏ ≤ π/π) c) π·(πΏ ≤ π/ππ) d) π·(πΏ > π/π) e) π·(πΏ ≤ π/π) Solution: - The sum of the probabilities is 1 and all probabilities are greater than or equal to zero. - P.m.f: f (1/8) = 0.2, f (1/4) = 0.7, f (3/8) = 0.1 a) π(π ≤ 1/8) = 0 b) π(π ≤ 1/4) = 0.9 c) π(π ≤ 5/16) = 0.9 d) π(π > 1/4) = 0.1 e) π(π ≤ 1/2) = 1 4. Suppose that π(π) = π−(π−π) πππ π > π. Determine the following probabilities: a) π·(πΏ > π) b) π·(π ≤ πΏ < π) c) π·(πΏ > π) d) π·(π < πΏ < ππ) e) Determine x, such that π·(πΏ < π) = π. π Solution: 5. Suppose the cumulative distribution function of the random variable X is: Determine the following: a) π·(πΏ < π. π) b) π·(πΏ > −π. π) c) π·(πΏ < −π) d) π·(−π < πΏ < π) Solution: 6. Suppose the probability density function of the length of computer cables is f(x) = 0.1 from 1200 to 1210 millimetres. a) Determine the mean and standard deviation of the cable length. b) If the length speciο¬cations are 1195 < x < 1205 millimetres, what proportion of cables are within speciο¬cations? Solution: 7. From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls. Solution: Denote by X the number of green balls in the three draws. Let G and B stand for the colors of green and black, respectively. The probability mass function for X is then: 8. Suppose a special type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The p.d.f that characterizes the proportion Y that make a profit is given by: π(π) = πππ (π − π)π , π ≤ π ≤ π and π(π) = π, πππππππππ a) What is the value of k that renders the above a valid density function? b) Find the probability that at most 50% of the firms make a profit in the first year. c) Find the probability that at least 80% of the firms make a profit in the first year. Solution: 1 a) Using integral by parts and setting π ∫0 π¦ 4 (1 − π¦)3 ππ¦ = 1, we obtain k = 280. b) For 0 ≤ π¦ ≤ 1, πΉ(π¦) = 56π¦ 5 (1 − π)3 + 28π¦ 6 (1 − π¦)2 + 8π¦ 7 (1 − π¦) + π¦ 8 So, π(π ≤ 5) = 0.3633 c) Using C.d.f in (b), π(π > 0.8) = 0.0563