How to determine cumulative probability, i.e. the probability of the variable X being less than µ + σ when we do not know X, µ or σ

Question

Let's say that you need to find the cumulative probability for a random variable X which is normally distributed.

You do not know what the value of X is or, for that matter, what the value of µ and σ is.

You only know that X = µ + σ.

Can you find the cumulative probability, i.e. the probability of the variable being less than µ + σ?

Answer 1

I assume you mean what is ${\Bbb P}(X \le \mu+\sigma)$ for $X$ a $N(\mu,\sigma)$-distributed random variable. The answer is the same as ${\Bbb P}(Y \le 1)$ for $Y$ a $N(0,1)$-distributed random variable, which is $\Phi(1)$ where $\Phi$ is the cumulative distribution function of a standard normal, and the answer is approximately $0.841345$.