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.
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