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 Jan16 asked Solution to Fredholm equation of the second type with symmetric Gaussian kernal Jun6 comment Is there a measure on $\mathbb{N}$? (What is the chance that a random integer from $\mathbb{N}$ is even?) @EricTowers, the counting measure is not a probability measure. Jun6 asked Counterexample for converse about measurable sections Jul7 comment solving a problem in probability Nitpicking: weight cannot be negative, hence the Gaussian assumption is unrealistic. Jun23 comment Mutually exclusive events Then you should assume $P(E) = P(F) = 0.5$ Jun22 comment Mutually exclusive events The event that $E$ does not occur first is (in my notaton) $A^c$. You cannot simply change the meaning of $E$ (which is an event in experiment $\mathcal E_1$). Perhaps the solution given by @DilipSarwate is close to what you are thinking: Think of the experiment in which either $E$ or $F$ occur for the first time. What is the probability that the event that occurred was $E$. Jun22 comment What is the best choice in a win/lose game and how to calculate it Agreed. To explain buying of lottery, you need to add additional utility for participating in the game---the thrill of the possibility of winning. Jun22 awarded Commentator Jun22 comment Mutually exclusive events Does my updated answer clarify this point? Jun22 comment What is the best choice in a win/lose game and how to calculate it (Tongue in cheek comment): Math doesn't help because math assumes linear utilities. Assume concave utilities, and math will explain risk adverse behavior and buying of lotteries. Jun22 awarded Editor Jun22 comment Mutually exclusive events If $P(E) = P(F) = 1$, then $E$ and $F$ cannot be mutually exclusive because $E \cup F \subset \Omega$, thus $P(E \cup F) = P(E) + P(F) \le P(\Omega) = 1$. Jun22 revised Mutually exclusive events More explanation Jun22 comment Mutually exclusive events $E^c = \{3,4,5,6\} \not\equiv \{3,4\} = F$ Jun22 answered Mutually exclusive events May17 comment Lower bound on a minimum of maximum of a sequence of standard normal random variables In the RHS of all equations, $j$ should be replaced by $p$. Apr26 comment Show that if $\{x_{n}\}$ and $\{y_{n}\}$ are Cauchy sequences in X then $d(x_{n},y_{n})$ converges in $\mathbb{R}$. In the reverse triangle inequality, the RHS should be $d(x,z)$. Oct23 comment Roots of $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ in the interval $[0,1]$ Yes, it is easy to see that the roots form a decreasing sequence. $f_{n+1}(x) - f_n(x) > 0$. If $a_n$ is a root of $f_n$ then, $f_{n+1}(a_n) > 0$. Thus, there is a root of $f_{n+1}$ between $0$ (where $f_{n+1}$ is negative) and $a_n$ (where $f_{n+1}$ is positive). Oct23 awarded Student Oct23 comment Roots of $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ in the interval $[0,1]$ @SL2:The derivative is not always positive. The derivative is $[ (n+2)x + 3n + 2 ](1-x)^n - 2(1-x)$ and its sign is determined by whether $a_n(x) = [(n+2)x + 3n + 2 ](1-x)^{n-1}$ is greater than or less than $2$. As $n \to \infty$, $a_n(x) \to 0$, for $x \in [0,1]$. Thus, the derivative, $a_n(x) - 2$, is negative for large $n$.