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].

• 0
• 0.2
• 0.5
• 0.8
• 1

References

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

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].