1. Consider X a uniform distribution over the interval [0, 1]. Now let Y = x/((x

Question

1. Consider X a uniform distribution over the interval [0, 1]. Now let Y = x/((x^2)+1). (a) (4%) Plot the function g(x) = x / (x2 + 1) over the interval (- , ). In particular, is g(x) an odd function or even function or neither (concepts of odd / even function are in exercise 3, Q2). Find the maximum of g(x). What is g()? You can plot by hand during in class exam, but have to use tool, program, or web site to show how g(x) looks in the redo final. (b) (2%) We know the PDF of X, f(x) = 1 in [0, 1], and f(x) = 0 outside this interval. Calculate the interval [a, b], so that the CDF of Y, FY(y) = 0 when y <= a, and FY(y) = 1 when y >= b. (c) (6%) Calculate FY(y), the CDF of Y.

Explanation / Answer

When X a uniform distribution over the interval [0, 1]. The maximum value of g(x) is 0.5 becuase x lies in between 0 to 1. The plot is when x range is 0 to 1.

x=runif(10000,0,1)
y=function(x){x/((x^2)+1)}
gx=y(x)
plot(x,gx)

When range is lies from -infinity to infinity then

curve(y,-10,10)

For g(infinity)=0 see in the graph below

curve(y,-1000,1000)

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