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Part 1:
The exponential distribution describes the time between independent events which occur continuously at a constant average rate. The probability distribution function of an exponential distribution is given by f(x) = \lambda e^{-\lambda x}. This is defined for x\geq 0 , where \lambda is some parameter of the distribution.
We first note that for larger values of \lambda , the gradient of the PDF is greater. Thus the parameter of the red curve, \lambda_{Red} is greater than the parameter of the blue curve, \lambda_{Blue}.
To find the value of the constant \lambda we can use boundary conditions.
At x=0 on the red curve, we can see that f(x) = f(0) = 2
\lambda e^{0} = \lambda = 2
f(x) =2e^{-2x}
And at x = 0 on the blue curve, we can see that f(x) = f(0) = 1
\lambda e^0 = \lambda = 1
f(x) =e^{-x}
Thus \lambda_{Red}=2 and \lambda_{Blue}=1, and \lambda_{Red}> \lambda_{Blue} as expected.
To find the mean of the exponential distribution we use the formula \bar{x}=\int xf(x) \,dx
By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.