Question

If Xi s a nonnegative random variable with mean 25, what can be said about:(a) E[X^3]?(b) E[√X]?(c) E[logX]?(d) E[e^−X]?

Answer 1

`E(x^3) \geq 15625`

`E(√(x)) \leq 5`

`E(\log x) \leq 1.3979`

`E(e^(-x)) \geq e^(-(25))`

Explanation:

We are given the following information in the question:

`X_i` s a nonnegative random variable with mean 25

`E(X) = 25`

We have to find the following:

a) Since
`f(x) = x^3` is a convex function,

`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`

b) Since
`f(x) = -√(x)` is a convex function,

`E(f(x))\geq f(E(x))\\\Rightarrow E(-√(x)) \geq -√((E(x)))\\\Rightarrow E(-√(x)) \geq -√((25))\\\Rightarrow E(-√(x)) \geq -5\\\Rightarrow E(√(x)) \leq 5`

c) Since
`f(x) = -\log x` is a convex function,

`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`

d) Since
`f(x) = e^(-x)` is a convex function,

`E(f(x))\geq f(E(x))\\\Rightarrow E(e^(-x)) \geq e^(-(E(x)))\\\Rightarrow E(e^(-x)) \geq e^(-(25))`