{
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Experiment with the <a href=https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.beta.html> scipy.stats.beta() </a> function for generating a beta function. Choose different parameters a and b and find values that give a significantly asymmetric probability distribution function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import beta\n",
    "a=   # choose some different values\n",
    "b=\n",
    "x=np.linspace(0,1,101)    # x values for plotting\n",
    "pdf=                      # use the method to get the pdf for your choices of a and b\n",
    "plt.plot(x,pdf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a function that will take as arguments beta function parameters and a number of points to draw a sample from. Your function should draw a sample of the requested size and return the mean of that sample."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def betamean(a,b,n) :\n",
    "    \"\"\" function to return the mean of n samples from a beta distribution with parameters a and b\n",
    "    \"\"\"\n",
    "    r =           # use the method to return n samples from a beta function with input a and b\n",
    "    return r.mean()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now write a loop that draws many samples of a given size, and plot a histogram of the mean values. Do this for several different sample sizes, and compare the resulting distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ndraws=               # choose the number of draws for each sample size to get a well sampled distribution\n",
    "for nsamp in [...] :  # add a list of different sample sizes here\n",
    "    # build up an array of means\n",
    "    means=[]\n",
    "    for i in range(ndraws):\n",
    "        means.append(betamean(a,b,nsamp))\n",
    "    means=np.array(means)\n",
    "    # plot the histogram of means for this sample size\n",
    "    plt.hist(means,histtype='step',label='{:d}'.format(nsamp),bins=30)\n",
    "# label the lines\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Discuss how your results relate to the central limit theorem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>ANSWER HERE</b> :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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