9. A line passing through the origin is described by the equation y = Mx, where...

9. A line passing through the origin is described by the equation y = Mx, where M greater than or equal to 0 is a random slope. Let (theta) be the angle between this line and the horizontal axis, in the right side of the plane. Suppose that theta is uniformly distributed between 0 and 90 degrees (0 and pi/2 radians). What, then, is the pdf of M? What is the expected value of M?

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Expert Solution

The slope of the given equation is defined as:

Where is the angle made by the line with the horizontal axis.

Also, we are given here that:

Therefore the CDF for would be defined as:

Now the CDF for M is computed here as:

Now using the CDF for , we get here:

Now this is the CDF for M, the PDF for M could be obtained by simply differentiating the above function as:

This is the required PDF for M here.

Now the expected value of M here is computed as:

Let L = m², then dL = 2m dm

The above value tends to infinity.

Therefore the expected value of M does not exist and tends to infinity.

orchestra answered 2 years ago

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