1-distributions.Rmd

---
title: "Excercises"
output: html_notebook
---

# Settings

R setup 

```{r}
knitr::opts_chunk$set(engine.path = list(
  python = "/usr/local/anaconda3/bin/python",
  julia = "/Applications/Julia-1.3.app/Contents/Resources/julia/bin/"
))
```

Python setup

```{python}
#import numpy as np ## numerical 
#import scipy as sp ## scientific 
import scipy.stats as stats
import pandas as pd ## data processing
```

Julia setup

```{julia}
using Distributions ## for distributions
using DataFrames
```

# Exercise 1

Assume that football player with success rate 0.4 shot 10 times on goal. Let $X$ be a random variable denoting number of successful scores.  Please find:

1. Distribution of $X$
2. Probability that football player score exactly 4 times ($P(X=4)$)
3. Probability that football player score at least 7 times ($P(X>=7) = 1- P(X <= 6)$)

## Solutions in R

```{r}
## Solution to 1
data.frame(trials = 0:10, p = dbinom(x = 0:10, size = 10, prob = 0.4))

## Solution to 2

dbinom(x = 4, size = 10, prob = 0.4)

## Solution to 3
1 - pbinom(q = 6, size = 10, prob = 0.4)
```

## Solutions in Python

```{python}
## Solution to 1
x = range(0,11)
p = [stats.binom.pmf(i, 10, 0.4) for i in x]
pd.DataFrame(data = {"x": x, "p": p})

## Solution to 2
stats.binom.pmf(4,10,0.4)

## Solution to 3
1 - stats.binom.cdf(6,10,0.4)
```


## Solutions in Julia

```{julia}
## Solution  to 1
hcat(0:10, [pdf(Binomial(10,0.4), i) for i in 0:10]) ## or hcat(0:10, pdf.(Binomial(10,0.5), 0:10))

## Solution to 2
pdf(Binomial(10,0.4), 4)

## Solution to 3
1 - cdf(Binomial(10,0.4), 6)
```

# Exercise 2

Number of car accidents in one day in some city follows Poisson distribution with expected value $\lambda=2$. Find the probability that at most 4 car accidents happen.

## Solution in R 

```{r}
ppois(q = 4, lambda = 2)
```

## Solution in Python

```{python}
stats.poisson.cdf(4, 2)
```

## Solution in Julia

```{julia}
cdf(Poisson(2), 4)
```