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    "# Return of Investments for online marketing\n",
    "\n",
    "## Try me\n",
    " [![Open In Colab](../../_static/colabs_badge.png)](https://colab.research.google.com/github/ffraile/operations-research-notebooks/blob/main/docs/source/Simulation/Exercises/Return%20of%20search%20engine%20investments.ipynb)[![Binder](../../_static/binder_badge.png)](https://mybinder.org/v2/gh/ffraile/operations-research-notebooks/main?labpath=docs%2Fsource%2FSimulation%2FExercises%2FReturn%20of%20search%20engine%20investments.ipynb)\n",
    "\n",
    "The following metrics and indicators determine the expenditures and return of investments of marketing investments in paid inclusion (including your company's web page in search results) for your company:\n",
    "\n",
    "- **S (Total Search results in period):** Total number of times the keywords are used in searches in the planning period.\n",
    "- **CPM (Cost per 1000 impressions):** Cost of the search engine per every 1000 impressions, every time your website appears in a search result\n",
    "- **CPC (Cost per click):** Cost per every click\n",
    "- **CTR (Click Through Rate):** rate between the number of clicks and number of impressions\n",
    "- **CVR (Conversion Rate):** rate between the number of visitors and the number of users\n",
    "\n",
    "With this, the number of visitors (V) and the number of users (U) can be calculated as:\n",
    "\n",
    "$U = V * CVR = S * CTR * CVR$\n",
    "\n",
    "The total costs ($C_t$) of the search engine investmets are:\n",
    "\n",
    "$C_t = CPC*V + CPM/1000*S$\n",
    "\n",
    "\n",
    "The **ARPU (Average revenue per user in planning period)** is the average revenue per user, or the total revenue in the planning period (R), divided by the number of users:\n",
    "\n",
    "$ARPU = R/U$\n",
    "\n",
    "The Return of Investments for online marketing is calculated as:\n",
    "\n",
    "$ROI = \\frac{R - C_t}{C_t}$\n",
    "\n",
    "After some research, you have found that the different metrics can be modeled as normal distributions with the following parameters:\n",
    "\n",
    "| Metric | Mean | Standard Deviation |\n",
    "|--------|------|--------------------|\n",
    "|Search results | 600000 | 100000 |\n",
    "| CTR | 0.05 | 0.01 |\n",
    "| CVR | 0.5 | 0.1 |\n",
    "\n",
    "These are the search engine fees for your keywords:\n",
    "- CPM = 0,3€\n",
    "- CPC = 0,75€\n",
    "\n",
    "You estimate your ARPU in 5€\n",
    "\n",
    "**a.** Calculate the total costs, the total Return of Investments and the average cost per acquisition using Montecarlo Simulation\n",
    "\n",
    "**b.** Calculate the 95% confidence interval of the ROI"
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