Prove or disprove: If f1(x) and f2(x) are probability density functions and if θ1 + θ2 = 1, then θ1f1(x) + θ2f2(x) is a probability density function.

I do not know how to do this at all. Any help would be greatly appreciated.

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- Jul 21st 2010, 04:04 AMcieloproving or disproving probabillity density function
Prove or disprove: If f1(x) and f2(x) are probability density functions and if θ1 + θ2 = 1, then θ1f1(x) + θ2f2(x) is a probability density function.

I do not know how to do this at all. Any help would be greatly appreciated. - Jul 21st 2010, 04:35 AMmr fantastic
1. $\displaystyle \int \theta_1 f_1(x) \, dx + \int \theta_2 f_2(x) \, dx = 1$ where the integrals are over the supports of each pdf.

But ....

2. Is $\displaystyle \theta_1 f_1(x) + \theta_2 f_2(x) \geq 0$ where $\displaystyle \theta_1 + \theta_2 = 1$ always true ....? - Jul 22nd 2010, 12:18 AMcielo
How nice! Thank you so much for the help!