SOLUTION: A discrete random variable X is such that P(X = 2^n)= 1/2^n , n = 1, 2,.... Show that EX = ∞. That is, X has no mathematical expectation Thank you

Question 1177400: A discrete random variable X is such that
P(X = 2^n)= 1/2^n , n = 1, 2,....
Show that EX = ∞. That is, X has no mathematical expectation
Thank you
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**1. Define the Expected Value**
The expected value of a discrete random variable X is defined as:
* E(X) = Σ [x * P(X = x)]
In our case, X takes values 2^n, and P(X = 2^n) = 1/2^n. So:
* E(X) = Σ [2^n * (1/2^n)] for n = 1, 2, 3, ...
**2. Simplify the Expression**
* E(X) = Σ [2^n / 2^n]
* E(X) = Σ [1] for n = 1, 2, 3, ...
**3. Analyze the Sum**
The sum is:
* E(X) = 1 + 1 + 1 + 1 + ...
This is an infinite sum of 1's.
**4. Conclusion**
Since we are adding 1 infinitely many times, the sum diverges to infinity.
* E(X) = ∞
Therefore, the expected value of X is infinite, meaning X has no mathematical expectation.

RELATED QUESTIONS
A discrete random variable X is such that P (X = 2n) = 1 / 2^n , n = 1, 2,.... Show... (answered by CPhill)

A discrete random variable X is such that P(X = n) = 2^n−1 / 3^n , n = 1,... (answered by CPhill)

A discrete random variable X is such that P(X = n) = 2^n−1 / 3^n , n = 1, 2, . . . ,... (answered by CPhill)

Suppose the random variable X has a binomial(n,U)distribution where U is uniformly... (answered by CPhill)

if p-n^2=x and (n+1)^2-p=y then,prove that p-xy is a perfect... (answered by robertb)

prove that (n 0) + (n 1) + (n 2) + ... + (n k) = 2^n is true using mathematical induction (answered by Shin123,ikleyn)

Show that (n + 1)^2 is a divisor of (n + 1)! + n! + (n -... (answered by tommyt3rd)

Suppose that X is a discrete random variable with probability mass function P(x)= cx^2 ,... (answered by ikleyn)

State the value of "n" such that ∠XYZ is a right angle. X(2, 1) Y(n, n) Z(2 + n,... (answered by Boreal)