Questions and Answers #9 Random Variables

Questions and Answers Sheet 9 Random Variables Question #1 What are the mean and standard deviation of a binomial probability distribution with n=23 and 𝑝 = 7 23 ? Answer: Explanation: Binomial distribution is a frequency distribution of possible number of successful outcomes in given number of trials n, in each of which there is the probability of success is p(and q=1-p is probability of failure). The mean and standard deviation of a binomial distribution are given by np and √𝑛𝑝𝑞. Here 𝑛 = 23, 𝑝 = 7 and 𝑞 = 1 − 23 Hence mean is 23 × 7 23 =√ 23 = 16 23 . =7 Standard deviation is √23 × 7×16 7 23 7 23 × 16 23 = √4.8696 = 2.21 Question #2 What are the mean and standard deviation of a binomial probability distribution with n=11 and 𝑝 = 7 30 Answer: Explanation: For a binomial probability distribution with n=11 and 𝑝 = Mean is given by 𝑛𝑝 = 11 × 7 30 7 30 = 2.567 Standard Deviation is given by √𝑛𝑝𝑞 where q=1-p Hence, Standard Deviation is √11 × = √11 × = 7 30 √11×7×23 30 × 7 30 × (1 − 7 30 ) 23 30 = 1.403 Question #3 What are the mean and standard deviation of a binomial probability distribution with n=8 and 𝑝 = Answer: 𝑚𝑒𝑎𝑛 = 56 29 standard deviation = 4√77 29 Explanation: Given Binomial Probability Distribution 𝑚𝑒𝑎𝑛 = 𝑛𝑝 = 8 ∗ 7 29 = 56 29 standard deviation = √𝑛𝑝(1 − 𝑝) = √8 ∗ 7 29 ∗ 22 29 = 4√77 29 7 29 ? ? Question #4 What are the mean and standard deviation of a binomial probability distribution with n=12 and 𝑝 = 7 60 ? Answer: Explanation: The mean and standard deviation of a binomial probability distribution are np and √𝑛𝑝𝑞 respectively where n is number of trials and p is probability of success of the event. Here q=1-p, and is the probability of failure of the event. 7 7 53 With n =12 and 𝑝 = , 𝑞 = 1 − = . 60 7 Hence mean is 12 × 60 60 7 5 Standard Deviation is √12 × = √4452 60 = 66.7233 60 60 = = 1.4 and 7 60 × 53 60 1 = ( ) √7 × 12 × 53 60 = 1.112 Question #5 What are the mean and standard deviation of a binomial probability distribution with n=100 and 𝑝 = 8 10 ? Answer: The Binomial Distribution B(n,P) is given by: n P( X = k ) = ( ) p k (1 − p)n−k k with 𝜇𝑋 = 𝑛𝑝 = 100 ∗ 8 10 = 80 8 2 10 10 𝜎𝑋2 = 𝑛𝑝(1 − 𝑝) = 100 ( ) ( ) = 16 Question #6 4 What are the mean and standard deviation of a binomial probability distribution with n=10 and 𝑝 = ? 5 Answer: Explanation: 4 𝑀𝑒𝑎𝑛 = 𝑛𝑝 = 10 × = 8 5 𝑆𝐷 = √𝑛𝑝𝑞 4 1 5 5 𝑞 =1−𝑝 =1− = 4 1 5 5 𝑆𝐷 = √10 × × = 1.265 Question #7 3 What are the mean and standard deviation of a binomial probability distribution with n=12 and 𝑝 = ? 5 Answer: Explanation: Mean of a binomial probability distribution is given by np and standard deviation is given by √𝑛𝑝𝑞, where q=1-p. 3 In the given example, 𝑛𝑝 = 12 ∗ = 7.2, 5 3 2 3 2 1 As 𝑞 = 1 − = , √𝑛𝑝𝑞 = √12 ∗ ∗ = ∗ √72 = 6 5 5 5 5 5 √2 5 = 1.697 Hence mean is 7.2 and standard deviation is 1.697 Question #8 3 What are the mean and standard deviation of a binomial probability distribution with n=130 and 𝑝 = ? 4 Answer: Mean is nxp and variance is nxpxq Explanation: Here n=130 3 𝑝= 𝑞= 4 1 4 3 So mean = 130 × ( ) = 390 4 = 97.5 4 3 1 390 4 4 16 Variance = 130 × × = = 24.375 Standard deviation = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = √24.375 Question #9 7 What are the mean and standard deviation of a binomial probability distribution with n=130 and 𝑝 = ? 9 Answer: Explanation: 7 𝑋 ∼ 𝐵 (130, ) 9 7 1 9 9 Mean = np, so 130 × = 101 Variance = npq. We know that p+q=1 2 So 𝑞 = 9 7 2 38 9 9 81 Therefore the variance is 130 × × = 22 Standard deviation is the square root of the variance, so 𝑠. 𝑑. = √22 38 81 = 4.74 Question #10 7 What are the mean and standard deviation of a binomial probability distribution with n=150 and 𝑝 = ? 9 Answer: mean =np standard deviation = √𝑛𝑝𝑞 Explanation: 7 350 mean = 150 × = ≈ 116.7 9 3 7 7 9 9 standard deviation = √150 × × (1 − ) ≈ 5.09