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  1. Let X be a continuous random variable with values exponentially distributed over [0,???) with parameter ?? = 0.1.
  1. Find the mean and variance of \(X\)
lambda <- 0.1
1/lambda #mean
## [1] 10
1/lambda^2 #variance
## [1] 100
  1. Using Chebyshev’s Inequality, find an upper bound for the following probabilities: \(P\left ( \left |X-10 \right |\geq 2 \right )\), \(P\left ( \left |X-10 \right |\geq 5 \right )\), \(P\left ( \left |X-10 \right |\geq 9 \right )\), \(P\left ( \left |X-10 \right |\geq 20 \right )\).
sigma_sq <- 1/lambda^2
sigma_sq / 2^2 #i
## [1] 25
sigma_sq / 5^2 #ii
## [1] 4
sigma_sq / 9^2 #iii
## [1] 1.234568
sigma_sq / 20^2 #iv
## [1] 0.25
  1. Calculate these probabilities exactly, and compare with the bounds in (b).
1-exp(-0.1*8) + exp(-0.1*12) #i
## [1] 0.8518652
1-exp(-0.1*5) + exp(-0.1*15) #ii
## [1] 0.6165995
1-exp(-0.1*1) + exp(-0.1*19) #iii
## [1] 0.2447312
1-exp(-0.1*0) + exp(-0.1*30) #iv
## [1] 0.04978707