Let X be a continuous random variable with the desnity function F(x).pdf

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Let X be a continuous random variable with the desnity function F(x)= 1/2 e^-|x| for all real numbers. Find the E(x) and Var(x). Solution Firstly defining F(x) i.e. F(x) = 0.5 * e^-x , x > = 0 = 0.5 * e^x , x < 0 therefore , for E(x) = ? x* F(x) * dx , limits from - infinity to infintiy putting F(x) and breaking limit at x = 0 E(x) = {? 0.5 *x* e^x + ? 0.5 *x* e^-x } dx for 1st part limits are - infinity and zero for 2nd par limits are zero and infinity on solving E (x) = zero For V(x) V(x) = ? x^2* F(x) * dx - E(x) on putting values and breaking limits at x = 0 V(x) = {? 0.5 *x^2* e^x + ? 0.5 *x^2* e^-x } dx - 0 for 1st part limits are - infinity and zero for 2nd par limits are zero and infinity on solving V(x) = 2 , E(x) = 0.

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$Let X be a continuous random variable with the desnity function F(x)= 1/2 e^-|x| for all real numbers. Find the E(x) and Var(x). Solution Firstly defining F(x) i.e. F(x) = 0.5 * e^-x , x > = 0 = 0.5 * e^x , x < 0 therefore , for E(x) = ? x* F(x) * dx , limits from - infinity to infintiy putting F(x) and breaking limit at x = 0 E(x) = {? 0.5 *x* e^x + ? 0.5 *x* e^-x } dx for 1st part limits are - infinity and zero for 2nd par limits are zero and infinity on solving E (x) = zero For V(x) V(x) = ? x^2* F(x) * dx - E(x) on putting values and breaking limits at x = 0 V(x) = {? 0.5 *x^2* e^x + ? 0.5 *x^2* e^-x } dx - 0 for 1st part limits are - infinity and zero for 2nd par limits are zero and infinity on solving V(x) = 2 , E(x) = 0$

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