Revision as of 19:04, 24 June 2024 by Admin (Created page with "Let <math>X</math> and <math>Y</math> be two random variables defined on the finite sample space <math>\Omega</math>. Assume that <math>X</math>, <math>Y</math>, <math>X + Y</math>, and <math>X - Y</math> all have the same distribution. Determine <math>P(X = Y = 0) </math>. '''References''' {{cite web |url=https://math.dartmouth.edu/~prob/prob/prob.pdf |title=Grinstead and Snell’s Introduction to Probability |last=Doyle |first=Peter G.|date=2006 |access-date=June 6...")
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ABy Admin
Jun 24'24

Exercise

Let [math]X[/math] and [math]Y[/math] be two random variables defined on the finite sample space [math]\Omega[/math]. Assume that [math]X[/math], [math]Y[/math], [math]X + Y[/math], and [math]X - Y[/math] all have the same distribution. Determine [math]P(X = Y = 0) [/math].

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

ABy Admin
Jun 25'24

Solution: E

We have

[[math]] E((X-Y)^2]=E[X^2] + E[Y^2] - 2E[XY]=E((X+Y)^2] = E[X^2] + E[Y^2] + 2E[XY]. [[/math]]

Hence [math]E[XY] = 0 [/math]. Then we also have

[[math]] E[X^2] = E[(X-Y)^2] = E[X^2] + E[Y^2] = 2E[X^2]. [[/math]]

Hence [math]E[X^2] = 0 [/math] which means that [math]P(X=0) = 1 [/math].

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