- #1
island-boy
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Once again, I'm having a disagreement with my TA regarding a problem set he gave us.
Here is the exact question, as written:
Find the distribution function associated with the following density functions:
a) f(x) = 3(1-x^2) , x an element of (0,1)
b)g(x) = x^{-2}, x an element of positive real numbers
here's where my problem lies.
For a)
if you solve for the cdf, you get:
F(x) = \int_{0}^{x} 3(1-t^2) dt = [3t - t^3]_{0}^{x} = 3x - x^3 for x element (0,1)
however, this value of F(x) takes the value of 2 when x = 1, which violates the property of a cdf! Also, if you take the integral of the density from 0 to 1, you will get 2! again a violation of the property of a density function, as the integral should be between 0 and 1.
for b)
if you solve for the cdf, you get
F(x) = \int_{0}^{x} t^{-2} dt = [-t^{-1}]_{0}^{x} = - 1/x + \infty for x element positive real
which again is greater than 1 for any value of x, positive real.
and if you get the integral of the density function from 0 to infinity (as the density function is defined for all positive real), you get infinity! which is not between 0 and 1.
the TA, however said that there is NOTHING wrong with the questions, even after he inspected it.
Am I insane to think that the questions are wrong? or am I not seeing something obvious?
help please. I'm going crazy.
Here is the exact question, as written:
Find the distribution function associated with the following density functions:
a) f(x) = 3(1-x^2) , x an element of (0,1)
b)g(x) = x^{-2}, x an element of positive real numbers
here's where my problem lies.
For a)
if you solve for the cdf, you get:
F(x) = \int_{0}^{x} 3(1-t^2) dt = [3t - t^3]_{0}^{x} = 3x - x^3 for x element (0,1)
however, this value of F(x) takes the value of 2 when x = 1, which violates the property of a cdf! Also, if you take the integral of the density from 0 to 1, you will get 2! again a violation of the property of a density function, as the integral should be between 0 and 1.
for b)
if you solve for the cdf, you get
F(x) = \int_{0}^{x} t^{-2} dt = [-t^{-1}]_{0}^{x} = - 1/x + \infty for x element positive real
which again is greater than 1 for any value of x, positive real.
and if you get the integral of the density function from 0 to infinity (as the density function is defined for all positive real), you get infinity! which is not between 0 and 1.
the TA, however said that there is NOTHING wrong with the questions, even after he inspected it.
Am I insane to think that the questions are wrong? or am I not seeing something obvious?
help please. I'm going crazy.
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