Last updated at May 29, 2018 by Teachoo

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Misc 4 Given that 𝑥 is the mean and σ2 is the variance of n observations 𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛 . Prove that the mean and variance of the observations 𝑎 𝑥1,𝑎 𝑥2,𝑎 𝑥3, ..., 𝑎𝑥𝑛 are a 𝑥 and a2σ2, respectively (a ≠ 0). Given observations are 𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛 and 𝑥 be their mean and σ2 is the variance Fro new observations, each observation is multiplied by a Let the new observations be 𝑦1, 𝑦2, 𝑦3, ..., 𝑦𝑛 where 𝑦𝑖 = a( 𝑥𝑖) Calculating new mean New mean = 1𝑛 𝑦𝑖 𝑦 = 1𝑛 𝑎𝑥𝑖 𝑦 = a × 1𝑛 𝑥𝑖 𝑦 = a 𝑥 So, New Mean = a 𝑥 Calculating new variance New Variance = 1n ( 𝑦𝑖− 𝑦)2 Now, Old Variance = 1𝑛 ( 𝑥𝑖− 𝑥)2 𝜎2 = 1𝑛 ( 𝑥𝑖− 𝑥)2 𝑛 𝜎2 = ( 𝑥𝑖− 𝑥)2 ( 𝑥𝑖− 𝑥)2 = 𝑛 𝜎2 ( 1𝑎 𝑦𝑖− 1𝑎 𝑦)2 = 𝑛 𝜎2 1𝑎2 (𝑦𝑖− 𝑦)2 = 𝑛 𝜎2 (𝑦𝑖− 𝑦)2 = 𝑛 𝜎2a2 So, New Variance = 1𝑛 ( 𝑦𝑖− 𝑦)2 = 1𝑛 × 𝑛 𝜎2a2 = 𝜎2a2 Hence, New mean = a 𝑥 & New variance = 𝜎2a2 Hence proved

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.