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<math> f_Y (y) = \frac{1}{3 \sqrt{2\pi} } e^{\frac{-(x-7)^2}{6}}.</math> | <math> f_Y (y) = \frac{1}{3 \sqrt{2\pi} } e^{\frac{-(x-7)^2}{6}}.</math> | ||
| − | Assuming that X and Y are independent, find the joint probability function <span class="texhtml">''f''<sub>''X''''Y'''''</sub>'''''('''''<b>x'',''y'').''</span> | + | Assuming that X and Y are independent, find the joint probability function <span class="texhtml">''f''<sub>''X''''Y'''''</sub>'''''('''''<b>x'',''y'').''</span> |
| + | </b></span> | ||
| + | ''' | ||
---- | ---- | ||
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Because X and Y are independent, the joint probability function can be represented as the product of the two marginal density functions:<br> | Because X and Y are independent, the joint probability function can be represented as the product of the two marginal density functions:<br> | ||
| − | <span class="texhtml" | + | <span class="texhtml"</span><math>f_{XY}(x,y) = f_X(x)f_Y(y)</math><br> |
Thus, the joint probability function is simply the two marginal density functions multiplied together: | Thus, the joint probability function is simply the two marginal density functions multiplied together: | ||
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| − | [[2013 Spring ECE 302 Boutin|Back to ECE302 Spring 2013 Prof. Boutin]] | + | [[2013 Spring ECE 302 Boutin|Back to ECE302 Spring 2013 Prof. Boutin]] |
| + | ''' | ||
| − | [[ECE302|Back to ECE302]] | + | '''[[ECE302|Back to ECE302]]''' |
[[Category:ECE302]] [[Category:ECE302Spring2013Boutin]] [[Category:Problem_solving]] [[Category:Continuous_random_variable]] | [[Category:ECE302]] [[Category:ECE302Spring2013Boutin]] [[Category:Problem_solving]] [[Category:Continuous_random_variable]] | ||
Revision as of 17:26, 1 March 2013
[[Category:independent random variables
Contents
Practice Problem: obtaining the joint pdf from the marginals of two independent variables
A random variable X has the following probability density function:
$ f_X (x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}}. $
Another random variable Y has the following probability density function:
$ f_Y (y) = \frac{1}{3 \sqrt{2\pi} } e^{\frac{-(x-7)^2}{6}}. $
Assuming that X and Y are independent, find the joint probability function fX'Y(x,y).</span>
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
Because X and Y are independent, the joint probability function can be represented as the product of the two marginal density functions:
<span class="texhtml"</span>$ f_{XY}(x,y) = f_X(x)f_Y(y) $
Thus, the joint probability function is simply the two marginal density functions multiplied together:
$ f_{XY}(x,y) = \frac{1}{6\pi} e^{\frac{1}{6}(-4x^2+14x-49)}. $
Answer 2
Write it here.
Answer 3
Write it here.
