3. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-∞, ∞) is a function f such that f(x) \geq 0 \int_{-\infty}^{\infty} f(x)=1 (a) Determine which of the following functions are probability density functions on the(-∞, ∞). \text { (i) } f(x)=\left\{\begin{array}{ll} x^{-1} & 0<x<e \\ 0 & \text { otherwise } \end{array}\right. \text { (ii) } f(x)=\left\{\begin{array}{ll} \frac{-2}{(x-\sqrt{2})^{3}} & 0<x<2 \sqrt{2} \\ 0 & \text { otherwise } \end{array}\right. \text { (iii) } f(x)=\left\{\begin{array}{ll} \lambda e^{\lambda x} & 0<x<\infty \\ 0 & \text { otherwise } \end{array}\right. (b) We can also use probability density functions to find the expected value of the outcomes of the event – if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. ſẵxƒ(x) dx yields the expected value for a density f(x) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.

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