A randomiser machine produces normally distributed random real numbers with fixed but unknown mean and variance....

 

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A randomiser machine produces normally distributed random real numbers with fixed but unknown mean and variance. These numbers are fed to a computer that outputs zero for numbers strictly between 1 and 1, 1 if the number is greater or equal to 1, and -1 if the number is less or equal to -1. In other words, the original normally distributed variable with mean μ and standard deviation σ, X~ converted into another random variable Y = g(X), where Find the following probabilities. g(x) = P(Y= = (a) Researchers have studied the output of the computer and determined that the cumulative distribution function of the output variable Y has the following representation. Fy (y) = P(Y = -1): -1/2): P(-1<Y < 1): P(X = -1): P(-1 ≤ x ≤ 1): == 0 -1, if x < -1, 0, if 1<x< 1, 1, if x ≥ 1. 6/23 0 0, if y< -1, 6 23' 3 16 23' 1, 10/23 if - 1 ≤ y ≤ 0, ~ N(μ, σ²), is if 0 ≤ y ≤ 1, if y≥ 1. Ⓡa A randomiser machine produces normally distributed random real numbers with fixed but unknown mean and variance. These numbers are fed to a computer that outputs zero for numbers strictly between 1 and 1, 1 if the number is greater or equal to 1, and -1 if the number is less or equal to -1. In other words, the original normally distributed variable with mean μ and standard deviation σ, X~ converted into another random variable Y = g(X), where Find the following probabilities. g(x) = P(Y= = (a) Researchers have studied the output of the computer and determined that the cumulative distribution function of the output variable Y has the following representation. Fy (y) = P(Y = -1): -1/2): P(-1<Y < 1): P(X = -1): P(-1 ≤ x ≤ 1): == 0 -1, if x < -1, 0, if 1<x< 1, 1, if x ≥ 1. 6/23 0 0, if y< -1, 6 23' 3 16 23' 1, 10/23 if - 1 ≤ y ≤ 0, ~ N(μ, σ²), is if 0 ≤ y ≤ 1, if y≥ 1. Ⓡa