Questions and Answers #11 Random Variables

Questions and Answers Sheet 11 Random Variables Question #1 What are the mean and standard deviation of a binomial probability distribution with n=25 and 𝑝 = 10 17 ? Answer: Explanation: The following are the equations for the mean and variance of the binomial distribution (taken from Wikipedia: Binomial Distribution ) 𝜇 = 𝑛∗𝑝 𝑉𝑎𝑟 = 𝑛 ∗ 𝑝 ∗ (1 − 𝑝) where n is the number of trials, and p is the probability of success in each trial. Standard Deviation is simply the square root of the variance: 𝜎 = √𝑉𝑎𝑟 = √𝑛 ∗ 𝑝 ∗ (1 − 𝑝) using the values from the question 10 250 𝜇 = 25 ∗ = ≅ 0.588 17 𝜎 = √25 ∗ 10 17 17 ∗ (1 − 10 1750 289 )=√ 289 ≅ 2.46 Question #2 What are the mean and standard deviation of a binomial probability distribution with n=69 and 𝑝 = 5 23 ? Answer: 𝑛 = 69, 𝑝 = 5/23 = 0.21 Expected value of X, Mean = 𝜇 = 𝑛𝑝 = 69 ∗ 0.21 = 15 Expectation or mean is: 𝜇 = 15 Std Dev: 𝜎 = √𝑛𝑝(1 − 𝑝) = √69 ∗ (0.21)(1 − 0.21) = √11.7 ≈ 3.42 Question #3 What are the mean and standard deviation of a binomial probability distribution with n=460 and 𝑝 = 16 23 ? Answer: Explanation: Mean of the Binomial Distribution is 16 Mean = 𝑛𝑝 = 460 × = 320 𝑞 =1−𝑝 =1− 16 23 = 23 7 23 𝑆𝑑 = √𝑛𝑝𝑞 = √460 × 16 23 × 7 23 = √97 9 23 = 9.87 Question #4 What are the mean and standard deviation of a binomial probability distribution with n=11 and 𝑝 = 19 23 ? Answer: Explanation: Given n=11 19 𝑝= 23 Mean = 𝑛𝑝 = 11 × 𝑞 =1−𝑝 =1− 19 23 19 =9 23 = 4 2 23 23 𝑆𝐷 = √𝑛𝑝𝑞 = √11 × 19 23 4 × 23 = √1.58 = 1.26 Question #5 What are the mean and standard deviation of a binomial probability distribution with n=12 and 𝑝 = 15 23 ? Answer: Explanation: Given n=12 15 𝑝= 23 q=1-p 𝑞 =1− 15 23 = 8 23 Mean = 𝑛𝑝 = 12 × 15 23 =7 19 23 15 8 382 𝑆𝐷 = √𝑛𝑝𝑞 = √12 × × = √2 = 1.65 23 23 529 Question #6 What are the mean and standard deviation of a binomial probability distribution with n=112 and 𝑝 = 15 37 ? Answer: Explanation: Given n=112 𝑝= 15 37 𝑞 =1−𝑝 =1− 15 37 = Mean = 𝑛𝑝 = 112 × 22 37 15 37 = 45.4 = 26.9 𝑆𝐷 = √𝑛𝑝𝑞 = √26.9 = 5.1 Question #7 What are the mean and standard deviation of a binomial probability distribution with n=12 and 𝑝 = Answer: Explanation: Mean = n*p 1 𝑆𝐷 = (𝑛 ∗ 𝑝 ∗ 𝑞)2 q=1-p 5 37 ? Mean = 12 ∗ 5 = 37 60 37 = 1.62 1 Standard deviation = (12 ∗ 5 37 ∗ 32 2 ) 37 = 1.18 Question #8 What are the mean and standard deviation of a binomial probability distribution with n=87 and 𝑝 = 5 37 ? Answer: Explanation: Given n=87 5 𝑝= 37 Mean 𝑥 = 𝑛𝑝 = 87 × 5 37 = 11 28 37 SD 𝜎 = √𝑛𝑝𝑞 5 𝑞 =1−𝑝 =1− SD 𝜎 = √87 × 37 5 37 = × 32 37 32 37 = √10.17 = 3.22 Question #9 What are the mean and standard deviation of a binomial probability distribution with n=94 and 𝑝 = 32 37 ? Answer: Explanation: Mean of a binomial probability distribution is given by np and standard deviation is given by √𝑛𝑝𝑞, where q=1-p. 32 In the given example, 𝑛𝑝 = 94 ∗ = 81.2 37 As 𝑞 = 1 − 32 37 = 5 37 , √𝑛𝑝𝑞 = √94 ∗ 32 37 ∗ 5 37 = 4 37 ∗ √940 = 3.3 Hence mean is 81.2 and standard deviation is 3.3 Question #10 4 What are the mean and standard deviation of a binomial probability distribution with n=5 and 𝑝 = ? 7 Answer: Explanation: Mean of a binomial probability distribution is given by np and standard deviation is given by √𝑛𝑝𝑞, where q=1-p. 4 20 In the given example, 𝑛𝑝 = 5 ∗ = = 2.857, 7 7 4 3 4 3 1 √15 7 7 7 7 7 7 As 𝑞 = 1 − = , √𝑛𝑝𝑞 = √5 ∗ ∗ = ∗ √60 = 2 = 1.1 Hence mean is 2.857 and standard deviation is 1.107