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   "source": [
    "# Introduction to mathematical statistics \n",
    "\n",
    "Welcome to lecture 3 in 02403\n",
    "\n",
    "During the lectures we will present both slides and notebooks. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: June 2024, Lotka-Volterra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = 10000\n",
    "alpha = 2/3\n",
    "beta = 4/3\n",
    "gamma = delta = 1\n",
    "sig_x = 1/8\n",
    "sig_y = 1/64\n",
    "mu_x = 1\n",
    "mu_y = 1/2\n",
    "x = stats.norm.rvs(size= k, loc = mu_x, scale = sig_x)\n",
    "y = stats.norm.rvs(size= k, loc = mu_y, scale = sig_y)\n",
    "K = y**alpha * np.exp(-beta * y) * x**gamma * np.exp(-delta * x)\n",
    "print(\"mu=\",np.mean(K))\n",
    "print(\"sigma^2=\",np.var(K, ddof = 1))\n",
    "print(\"sigma=\",np.std(K,ddof = 1))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(K,density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## June 2024 part two"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu_x = np.arange(0.5,2,0.01)\n",
    "K = mu_y**alpha * np.exp(-beta * mu_y) * mu_x**gamma * np.exp(-delta * mu_x)\n",
    "Kx = K * (gamma / mu_x - delta)\n",
    "Ky = K * (alpha / mu_y - beta)\n",
    "Vk = Kx**2 * sig_x**2 +Ky**2 * sig_y**2\n",
    "fig, ax =plt.subplots(2,1)\n",
    "ax[0].plot(mu_x,K) ## mean\n",
    "ax[1].plot(mu_x,Vk) ## mean\n",
    "plt.tight_layout()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[1/2,-1/2],[-1/2,1/2]])\n",
    "print(A)\n",
    "print(A @ A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "At = np.array([[1/2,-1/2],[-1/2,1/2],[1/2,1/2]])\n",
    "print(At)\n",
    "print(At @ At.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variance of normal sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 5\n",
    "k = 10000\n",
    "y = stats.norm.rvs(loc=10, scale=np.sqrt(2), size=(k,n))\n",
    "print(y)\n",
    "S2 = y.var(axis=1,ddof=1)\n",
    "plt.hist(S2,bins=20,density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: Meen and median"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 5\n",
    "k = 10000\n",
    "y = stats.norm.rvs(loc=10, scale=np.sqrt(2), size=(k,n))\n",
    "y_bar = y.mean(axis=1)\n",
    "y_tilde = np.median(y,axis=1)\n",
    "print(\"E[y_bar]\", np.mean(y_bar))\n",
    "print(\"E[y_tilde]\", np.median(y_bar))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"V[y_bar]\", np.var(y_bar,ddof=1))\n",
    "print(\"V[y_tilde]\", np.var(y_tilde,ddof=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example P(Q>10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "1-stats.chi2.cdf(10,df=10)"
   ]
  }
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