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Variance and Standard Deviation of a Random Variable
Variance and Standard Deviation of a Random Variable
Last updated at Feb. 10, 2020 by Teachoo
Ex 13.4, 15 In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var (X). Given that X = 0 is Members oppose X = 1 is members favour proposal Given, 70% of members favour proposal So, P(X = 1) = 70% = 0.7 and 30% of members oppose proposal So, P(X = 0) = 30% = 0.3 ∴ Probability distribution is The expectation value E(x) is given by : E 𝑿 = 𝑖 = 1𝑛𝑥𝑖𝑝𝑖 = 0 × 0.3 + 1 × 0.7 = 0.7 The variance of x is given by : Var 𝑋=𝐸 𝑋2− 𝐸 𝑋2 So, finding 𝐸 𝑋2 E 𝑿𝟐= 𝑖 = 1𝑛 𝑥𝑖2𝑝𝑖 = 02 × 0.3 + 12 × 0.7 = 0 + 0.7 = 0.7 Now, Var 𝒙 = 𝐸 𝑥2− 𝐸 𝑥2 = 0.7 – 0.72 = 0.7 1−0.7 = 0.7 0.3 = 0.21 Hence the expectation E(x) = 0.7 & variance var(x) = 0.21