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The dual iswhich is in standard maximization form after adding nonnegative slack variables. The simplex pivots first introduce the variable with coefficient seven and then the remaining payoff column; the optimum occurs where both constraints bind:Thus $y=(1/16,3/16)$ and the dual value is $1/4$. Complementary slackness gives $x=(1/8,1/8)$. Hence $v=4$ and the normalized optimal strategies are

Every payoff is positive, so every pair of mixed strategies has positive expected payoff and the game value satisfies $v>0$. Player 1 maximizes $v$ subject to $A^{T}p\ge ve$, $e^{T}p=1$, $p\ge 0$. Setting $x=p/v$ turns these intoMaximizing $v$ is therefore equivalent to the stated minimization of $e^{T}x$.

A pure optimum is a saddle point: an entry minimal in its row and maximal in its column. In $A_1$, $c$ at $(2,1)$ is a saddle; in $A_2$, $c$ at $(2,2)$ is a saddle. Matrix $A_3$ has row maximin $b$ and column minimax $c$, so no saddle exists. Thus

A two-player zero-sum game has a payoff gained by one player and lost by the other. A pure strategy chooses one action deterministically; a mixed strategy is a probability distribution over pure actions.

The null value $\pi =1/2$ is interior and the model is regular. Wilks theorem therefore givesunder the null, because the alternative adds one free parameter. An asymptotic size-$\alpha$ test rejects when

For known ${\lambda }_0$, the likelihood ratio of $\pi =1/2$ to $\pi =0$ iswhich increases with $n_0$. Reject for $n_0>c$, with randomization at $n_0=c$ chosen so that the null rejection probability is exactly $\alpha$. Since the zero indicators form a Bernoulli family with success probability increasing in $\pi$ and have a monotone likelihood ratio in $n_0$, the Karlin-Rubin theorem makes this test uniformly most powerful for $\pi >0$.

A statistic is sufficient when the conditional law of the sample given it is parameter-free; it is minimal sufficient when it is a function of every sufficient statistic. The likelihood factors asso the factorization theorem makes $(n_0,S)$ sufficient. The likelihood-ratio criterion shows that the ratio for two samples is parameter-independent exactly when both $n_0$ and $S$ agree. Thus

The Time-dependent Schrödinger equation is $i\hslash {\partial }_t\psi =\hat{H}\psi$. Differentiate $⟨\hat{A}⟩=⟨\psi |\hat{A}|\psi ⟩$, use the equation and its adjoint, and include explicit time dependence of the operator. This gives Ehrenfest theorem

The standard contour transformholds for $\mathrm{Re}\alpha >0$, with the appropriate upper or lower pole chosen according to the sign of $k$. Therefore

Integration by parts, using decay of $m$, gives the Fourier transform of a derivative

The Euler-Lagrange equation isPut $A=\sqrt{1+\sqrt{2}}$, $B=\sqrt{\sqrt{2}-1}$, and $L=\pi /2$. The conditions at zero reduce the general solution toDefineThe remaining boundary conditions givewhich determines the unique extremal explicitly.

DifferentiateThe product rule cancels all terms exceptOn an extremal with no explicit $x$ dependence both terms vanish, proving the stated conserved quantity.

For variations $y+\epsilon \eta$ with $\eta ={\eta }^{\prime }=0$ at both endpoints, integrating the ${\eta }^{\prime }$ term once and the ${\eta }^{\prime \prime }$ term twice givesThe fundamental lemma yields the higher-order Euler-Lagrange equation

The first three initial values vanish and the third derivative at zero equals one. Transforming the differential equation givesso $F(s)=1/[s^2(s+2{)}^2]$. Hence

The transform factors as $s^{-2}(s+2{)}^{-2}$, whose inverse factors are $t$ and $t{e}^{-2t}$. The Laplace convolution theorem givesand direct integration producesconfirming part (b).