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

Question 1177294: A discrete random variable X is such that
P (X = 2n) = 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!
You're right, let's break down why this discrete random variable X has an infinite expected value.
**Understanding the Problem**
* We're given a discrete random variable X.
* The possible values of X are 2^n, where n = 1, 2, 3, ... (i.e., 2, 4, 8, 16, ...).
* The probability of X taking the value 2^n is P(X = 2^n) = 1 / 2^n.
* We need to show that the expected value E(X) is infinite.
**Calculating the Expected Value**
The expected value of a discrete random variable is calculated as:
* E(X) = Σ [x * P(X = x)]
In our case:
* E(X) = Σ [2^n * (1 / 2^n)] for n = 1, 2, 3, ...
**Simplifying the Expression**
* E(X) = Σ [2^n / 2^n]
* E(X) = Σ [1] for n = 1, 2, 3, ...
**Analyzing the Sum**
This means:
* E(X) = 1 + 1 + 1 + 1 + ... (an infinite sum of 1s)
**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.**
**Regarding the Code and its output**
The provided code calculates the partial sum of the series, not the correct expected value. The code calculates the partial sum of Σ n * (2^(n-1)) / (2^n) and not Σ 2^n * (1/2^n). The correct expected value calculation, as shown above, results in an infinite sum.

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