Misc 4 - Given that x is mean and is variance of n observations - Miscellaneous

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  1. Chapter 15 Class 11 Statistics
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Misc 4 Given that 𝑥﷯ is the mean and σ2 is the variance of n observations 𝑥﷮1﷯, 𝑥﷮2﷯, 𝑥﷮3﷯, ..., 𝑥﷮𝑛﷯ . Prove that the mean and variance of the observations 𝑎 𝑥﷮1﷯,𝑎 𝑥﷮2﷯,𝑎 𝑥﷮3﷯, ..., 𝑎𝑥﷮𝑛﷯ are a 𝑥﷯ and a2σ2, respectively (a ≠ 0). Given observations are 𝑥﷮1﷯, 𝑥﷮2﷯, 𝑥﷮3﷯, ..., 𝑥﷮𝑛﷯ and 𝑥﷯ be their mean and σ2 is the variance Fro new observations, each observation is multiplied by a Let the new observations be 𝑦﷮1﷯, 𝑦﷮2﷯, 𝑦﷮3﷯, ..., 𝑦﷮𝑛﷯ where 𝑦﷮𝑖﷯ = a( 𝑥﷮𝑖﷯) Calculating new mean New mean = 1﷮𝑛﷯ ﷮﷮ 𝑦﷮𝑖﷯﷯ 𝑦﷯ = 1﷮𝑛﷯ ﷮﷮ 𝑎𝑥﷮𝑖﷯﷯ 𝑦﷯ = a × 1﷮𝑛﷯ ﷮﷮ 𝑥﷮𝑖﷯﷯ 𝑦﷯ = a 𝑥﷯ So, New Mean = a 𝑥﷯ Calculating new variance New Variance = 1﷮n﷯ ﷮﷮( 𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ Now, Old Variance = 1﷮𝑛﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ 𝜎﷮2﷯ = 1﷮𝑛﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ 𝑛 𝜎﷮2﷯ = ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ ﷮﷮( 𝑥﷮𝑖﷯﷯− 𝑥﷯)﷮2﷯ = 𝑛 𝜎﷮2﷯ ﷮﷮( 1﷮𝑎﷯ 𝑦﷮𝑖﷯﷯− 1﷮𝑎﷯ 𝑦﷯)﷮2﷯ = 𝑛 𝜎﷮2﷯ ﷮﷮ 1﷮𝑎﷯﷯﷮2﷯ (𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ = 𝑛 𝜎﷮2﷯ ﷮﷮ (𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ = 𝑛 𝜎﷮2﷯a2 So, New Variance = 1﷮𝑛﷯ ﷮﷮( 𝑦﷮𝑖﷯﷯− 𝑦﷯)﷮2﷯ = 1﷮𝑛﷯ × 𝑛 𝜎﷮2﷯a2 = 𝜎﷮2﷯a2 Hence, New mean = a 𝑥﷯ & New variance = 𝜎﷮2﷯a2 Hence proved

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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.