Revision as of 02:01, 28 June 2024 by Admin (Created page with "Find <math>E(X^Y)</math>, where <math>X</math> and <math>Y</math> are independent random variables which are uniform on <math>[0, 1]</math>. <ul class="mw-excansopts"> <li>0.6931</li> <li>0.7131</li> <li>0.7344</li> <li>0.7544</li> <li>0.775</li> </ul> '''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, 2024}}")
Jun 28'24
Exercise
Find [math]E(X^Y)[/math], where [math]X[/math] and [math]Y[/math] are independent random variables which are uniform on [math][0, 1][/math].
- 0.6931
- 0.7131
- 0.7344
- 0.7544
- 0.775
References
Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.
Jun 28'24
Solution: A
For fixed [math]y \in [0,1][/math], we have [math]E[X^y] = \frac{1}{y+1}[/math] and thus
[[math]]
E[X^Y] = \int_{0}^1 \frac{1}{y+1} \, dy = \log(2).
[[/math]]