CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z

In summary: Please let us know what you've tried, and we'll be glad to help you from there. Thanks.In summary, the conversation discusses the derivation and sketching of the CDF for the distance from the origin (D), the CDF for the random angle between the origin and the point (A), and the probability P(Y > cX) for a constant c. The tools used include joint distributions and double integrals, but the exact method is not specified as it may vary depending on the individual's understanding and approach to the problem.
  • #1
london
3
0
Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y).

(a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d.

(b) The ratio T = Y/X has Student’s t-distribution with 1 degree of freedom, also called the Cauchy distribution with CDF FT (t) = P(T ≤ t) = 1/2 + 1/π tan−1(t). Use this to determine the CDF FA(a) = P(A ≤ a) of the random angle A = tan−1(Y/X) between the line joining the origin and (X, Y) and the X-axis, for −π/2 < a < π/2 (points below the X-axis subtend a negative angle). Sketch this CDF as a function of all real a.

(c) Determine the probability P(Y > cX) for a constant c.
 
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  • #2
london said:
Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y).

(a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d.

(b) The ratio T = Y/X has Student’s t-distribution with 1 degree of freedom, also called the Cauchy distribution with CDF FT (t) = P(T ≤ t) = 1/2 + 1/π tan−1(t). Use this to determine the CDF FA(a) = P(A ≤ a) of the random angle A = tan−1(Y/X) between the line joining the origin and (X, Y) and the X-axis, for −π/2 < a < π/2 (points below the X-axis subtend a negative angle). Sketch this CDF as a function of all real a.

(c) Determine the probability P(Y > cX) for a constant c.

Show your work.
 
  • #3
Ray Vickson said:
Show your work.

well i really have no idea where to start :/
But i was thinking maybe of using joint distributions but that leads to double integrals and i have not been taught double integrals so there must be another way about it right?
 
  • #4
london said:
well i really have no idea where to start :/
But i was thinking maybe of using joint distributions but that leads to double integrals and i have not been taught double integrals so there must be another way about it right?
Yes, it involves a double integral, but that should not stop you at least writing out the expression for it.
 
  • #5
haruspex said:
Yes, it involves a double integral, but that should not stop you at least writing out the expression for it.

but what's the point of writing it out when its supposed to be derived so I can sketch it? My lecturer said double integrals are not assessable so clearly another method is used?
 
  • #6
london said:
but what's the point of writing it out when its supposed to be derived so I can sketch it? My lecturer said double integrals are not assessable so clearly another method is used?

You need to do some work here. If there ARE some tools/results you are allowed to use you need to tell us about them. We cannot possibly help if we have zero information.

I urge you to read Vela's 'pinned' thread "Guidelines for students and helpers', which explains what the expectations are when you post to this Forum. In particular, stating that you have no idea how to start is not acceptable.
 
  • #7
london said:
Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y).

(a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d.

(b) The ratio T = Y/X has Student’s t-distribution with 1 degree of freedom, also called the Cauchy distribution with CDF FT (t) = P(T ≤ t) = 1/2 + 1/π tan−1(t). Use this to determine the CDF FA(a) = P(A ≤ a) of the random angle A = tan−1(Y/X) between the line joining the origin and (X, Y) and the X-axis, for −π/2 < a < π/2 (points below the X-axis subtend a negative angle). Sketch this CDF as a function of all real a.

(c) Determine the probability P(Y > cX) for a constant c.

Please check your PMs. Per the PF rules (see Site Info at the top of the page), you *must* show your efforts toward solving the problem before we can offer tutorial help.
 

FAQ: CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z

1. What is the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z?

The CDF (cumulative distribution function) of Distance and Angle from Origin of N(0, 1) RVs (random variables) Y and Z can be calculated using the standard normal distribution formula. This formula takes into account the mean (0) and standard deviation (1) of the normal distribution and uses integration to determine the probability of the random variables falling within a certain range of values.

2. How is the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z used in statistics?

The CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z is used to determine the likelihood of a random variable falling within a specific range of values. This is important in statistics because it allows us to make predictions and draw conclusions about a population based on a sample of data.

3. Can the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z be graphed?

Yes, the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z can be graphed using a graphing calculator or statistical software. The resulting graph will show the probability of the random variables falling within a certain range of values. The x-axis represents the values of the random variables and the y-axis represents the probability.

4. How does the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z compare to other probability distributions?

The CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z follows a normal distribution, which is a commonly used probability distribution in statistics. The shape of the graph is bell-shaped and symmetrical, with the majority of the data falling within one standard deviation of the mean. This makes it a useful tool for analyzing and making predictions about many different types of data.

5. What is the relationship between the CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z and the PDF (probability density function)?

The CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z is the integral of the PDF, which represents the probability distribution of the random variables. The PDF gives the probability of each individual value occurring, while the CDF shows the cumulative probability of all values up to a certain point. Therefore, the CDF and PDF are closely related and provide complementary information about the distribution of the random variables.

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