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   "source": [
    "The Binomial distribution\n",
    "visualize the Binomial distribution X~B(n,p)\n",
    "think of n coin tosses, sum number of \"face\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(repr.plot.width=10, repr.plot.height=6.5)   # this command just formats the size of the figures. Adapt to view them nicely\n",
    "                                                    # in your browser."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 50\n",
    "p <- 0.80\n",
    "\n",
    "k <- seq(0,n,by=1)                       #Binomial: x can take any value k between 0 and n.\n",
    "f_k <- dbinom(x = k,size = n,prob = p)   # compute pdf of B(n,k) with dbinom\n",
    "plot(k,f_k, ylab = \"PDF\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Show numerical pdf\n",
    "\n",
    "f_k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Normal distribution\n",
    "\n",
    "mean <- 5 # mu in literature\n",
    "sd <- 0.8 # sigma\n",
    "\n",
    "x <- seq(-5,10,length=1000)\n",
    "y <- dnorm(x,mean,sd)  #dnorm  : \"D\"ensity of \"NORM\"al distribution. This gives the PDF.\n",
    "\n",
    "plot(x, y, type=\"n\", xlab=\"Parameter X\", ylab=\"\",\n",
    "     main=\"Normal Distribution\", axes=FALSE)\n",
    "axis(1, at=seq(0, 10, 2), pos=0)\n",
    "lines(x, y)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Normal distribution\n",
    "mean=100\n",
    "sd=15\n",
    "lb=80      # lower bound of X\n",
    "ub=120     # upper bound of x   \n",
    "#lb=mean-1*sd; ub=mean+1*sd\n",
    "# This will compute the probability for lb < X < ub\n",
    "\n",
    "\n",
    "x <- seq(-4,4,length=100)*sd + mean\n",
    "hx <- dnorm(x,mean,sd)  #dnorm  : \"D\"ensity of \"NORM\"al distribution\n",
    "\n",
    "plot(x, hx, type=\"n\", xlab=\"Parameter X\", ylab=\"\",\n",
    "     main=\"Normal Distribution\", axes=FALSE)\n",
    "\n",
    "i <- x >= lb & x <= ub\n",
    "lines(x, hx)\n",
    "polygon(c(lb,x[i],ub), c(0,hx[i],0), col=\"red\")\n",
    "\n",
    "area <- pnorm(ub, mean, sd) - pnorm(lb, mean, sd)   #pnorm : \"P\"robability of \"NORM\" (=Phi(Z))\n",
    "result <- paste(\"P(\",lb,\"< X <\",ub,\") =\",\n",
    "                signif(area, digits=4))\n",
    "mtext(result,3)\n",
    "axis(1, at=seq(40, 160, 20), pos=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The Whiskey case\n",
    "\n",
    "1-pnorm(q = 69.5,mean = 67.5,sd = 2.5) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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