Respuesta :
Answer:
[tex]E(x^3) \geq 15625[/tex]
[tex]E(\sqrt{x}) \leq 5[/tex]
[tex]E(\log x) \leq 1.3979[/tex]
[tex]E(e^{-x}) \geq e^{-(25)}[/tex]
Step-by-step explanation:
We are given the following information in the question:
[tex]X_i[/tex] s a nonnegative random variable with mean 25
[tex]E(X) = 25[/tex]
We have to find the following:
a) Since [tex]f(x) = x^3[/tex] is a convex function,
[tex]E(f(x))\geq f(E(x))\\\Rightarrow E(x^3) \geq (E(x))^3\\\Rightarrow E(x^3) \geq (25)^3\\\Rightarrow E(x^3) \geq 15625[/tex]
b) Since [tex]f(x) = -\sqrt{x}[/tex] is a convex function,
[tex]E(f(x))\geq f(E(x))\\\Rightarrow E(-\sqrt{x}) \geq -\sqrt{(E(x))}\\\Rightarrow E(-\sqrt{x}) \geq -\sqrt{(25)}\\\Rightarrow E(-\sqrt{x}) \geq -5\\\Rightarrow E(\sqrt{x}) \leq 5[/tex]
c) Since [tex]f(x) = -\log x[/tex] is a convex function,
[tex]E(f(x))\geq f(E(x))\\\Rightarrow E(-\log x) \geq -\log (E(x))\\\Rightarrow E(-\log x) \geq -\log (25)^3\\\Rightarrow E(\log x) \leq 1.3979[/tex]
d) Since [tex]f(x) = e^{-x}[/tex] is a convex function,
[tex]E(f(x))\geq f(E(x))\\\Rightarrow E(e^{-x}) \geq e^{-(E(x))}\\\Rightarrow E(e^{-x}) \geq e^{-(25)}[/tex]
