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Learn how to describe random variables using probability distributions.

Introduction to Probability Distributions

We can take the difference between two overlapping ranges to calculate the probability that a random selection will be within a range of values for continuous distributions. This is essentially the same process as calculating the probability of a range of values for discrete distributions.

![Gif with two overlapping densities, subtract one out to find the difference, and therefore the probability in that range](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/Normal-PDF-Range.gif)

Let's say we wanted to calculate the probability of randomly observing a woman between 165 cm to 175 cm, assuming heights still follow the Normal(167.74, 8) distribution. We can calculate the probability of observing these values or less. The difference between these two probabilities will be the probability of randomly observing a woman in this given range. This can be done in python using the `norm.cdf()` method from the `scipy.stats` library. As mentioned before, this method takes on 3 values:
- `x`: the value of interest
- `loc`: the mean of the probability distribution
- `scale`: the standard deviation of the probability distribution

```python
import scipy.stats as stats
# P(165 < X < 175) = P(X < 175) - P(X < 165)
# stats.norm.cdf(x, loc, scale) - stats.norm.cdf(x, loc, scale)
print(stats.norm.cdf(175, 167.74, 8) - stats.norm.cdf(165, 167.74, 8))
```
Output:
```
# 0.45194
```

We can also calculate the probability of randomly observing a value or greater by subtracting the probability of observing less than the given value from 1. This is possible because we know that the total area under the curve is 1, so the probability of observing something greater than a value is 1 minus the probability of observing something less than the given value.

Let's say we wanted to calculate the probability of observing a woman taller than 172 centimeters, assuming heights still follow the Normal(167.74, 8) distribution. We can think of this as the opposite of observing a woman shorter than 172 centimeters. We can visualize it this way:

![Image showing how P(X > 172) = 1 - P(X < 172)](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/norm_pdf_167_8_filled2.svg)

We can use the following code to calculate the blue area by taking 1 minus the red area:

```python
import scipy.stats as stats

# P(X > 172) = 1 - P(X < 172)
# 1 - stats.norm.cdf(x, loc, scale)
print(1 - stats.norm.cdf(172, 167.74, 8))
```
Output:
```
# 0.29718
```

Probability Density Functions and Cumulative Distribution Function

Congrats! We have finished our introduction to probability distributions! To review, we have: 
* Learned about different types of random variables
* Calculated the probability of specific events using probability mass functions (discrete random variable)
* Calculated the probability of ranges using probability mass functions and cumulative distribution functions (discrete random variable)
* Calculated the probability of ranges using probability density functions and cumulative distribution functions (continuous random variable)

Review

A *probability mass function (PMF)* is a type of *probability distribution* that defines the probability of observing a particular value of a discrete random variable. For example, a PMF can be used to calculate the probability of rolling a three on a fair six-sided die.

There are certain kinds of random variables (and associated probability distributions) that are relevant for many different kinds of problems. These commonly used probability distributions have names and *parameters* that make them adaptable for different situations.

For example, suppose that we flip a fair coin some number of times and count the number of heads. The probability mass function that describes the likelihood of each possible outcome (eg., 0 heads, 1 head, 2 heads, etc.) is called the _binomial distribution_. The parameters for the binomial distribution are:

- `n` for the number of trials (eg., n=10 if we flip a coin 10 times)
- `p` for the probability of success in each trial (probability of observing a particular outcome in each trial. In this example, p= 0.5 because the probability of observing heads on a fair coin flip is 0.5) 

If we flip a fair coin 10 times, we say that the number of observed heads follows a `Binomial(n=10, p=0.5)` distribution. The graph below shows the probability mass function for this experiment. The heights of the bars represent the probability of observing each possible outcome as calculated by the PMF.

![A histogram with markers 0 to 10 along the x-axis and the heights of the bars at each marker represent the probability of observing the value of the marker from this distribution](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/binom_pmf_10_5.svg)


Probability Mass Functions

We can use the `binom.cdf()` method from the `scipy.stats` library to calculate the cumulative distribution function. This method takes 3 values:
- `x`: the value of interest, looking for the probability of this value or less
- `n`: the sample size
- `p`: the probability of success

Calculating the probability of observing 6 or fewer heads from 10 fair coin flips (0 to 6 heads) mathematically looks like the following:

```tex
P(6\; or\; fewer\; heads) = P(0\; to\; 6\; heads)
```

In python, we use the following code:

```python
import scipy.stats as stats

print(stats.binom.cdf(6, 10, 0.5))
```
Output:
```
0.828125
```

Calculating the probability of observing between 4 and 8 heads from 10 fair coin flips can be thought of as taking the difference of the value of the cumulative distribution function at 8 from the cumulative distribution function at 3:

```tex
P(4\;to\;8\;Heads) = P(0\;to\;8\;Heads) - P(0\;to\;3\;Heads)
```
In python, we use the following code:

```python
import scipy.stats as stats

print(stats.binom.cdf(8, 10, 0.5) - stats.binom.cdf(3, 10, 0.5))
```
Output:
```
# 0.81738
```

To calculate the probability of observing more than 6 heads from 10 fair coin flips we subtract the value of the cumulative distribution function at 6 from 1. Mathematically, this looks like the following:

```tex
P(more\; than\; 6\; heads) = 1 - P(6\; or\; fewer\; heads)
```

Note that "more than 6 heads" does not include 6. In python, we would calculate this probability using the following code:

```python
import scipy.stats as stats
print(1 - stats.binom.cdf(6, 10, 0.5))
```
Output:
```
# 0.171875
```

Using the Cumulative Distribution Function in Python

We have seen that we can calculate the probability of observing a specific value using a probability mass function. What if we want to find the probability of observing a range of values for a discrete random variable? One way we could do this is by adding up the probability of each value.

For example, let's say we flip a fair coin 5 times, and want to know the probability of getting between 1 and 3 heads. We can visualize this scenario with the probability mass function:

![GIF of highlighting selected bars on a histogram](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/Binomial-Distribution-PMF-Probability-over-a-Range.gif)


We can calculate this using the following equation where *P(x)* is the probability of observing the number *x* successes (heads in this case):

```tex
\begin{aligned}
P(1\;to\;3\;heads) = P(1 <= X <= 3) \\
P(1\;to\;3\;heads) = P(X=1) + P(X=2) + P(X=3) \\
P(1\;to\;3\;heads) = 0.1562 + 0.3125 + 0.3125 \\
P(1\;to\;3\;heads) = 0.7812 
\end{aligned}
```

Using the Probability Mass Function Over a Range

##### Discrete Random Variables

Random variables with a countable number of possible values are called _discrete random variables_. For example, rolling a regular 6-sided die would be considered a discrete random variable because the outcome options are limited to the numbers on the die. 

Discrete random variables are also common when observing counting events, such as how many people entered a store on a randomly selected day. In this case, the values are countable in that they are limited to whole numbers (you can't observe half of a person).


##### Continuous Random Variables

When the possible values of a random variable are uncountable, it is called a _continuous random variable_. These are generally measurement variables and are uncountable because measurements can always be more precise -- meters, centimeters, millimeters, etc. 

For example, the temperature in Los Angeles on a randomly chosen day is a continuous random variable. We can always be more precise about the temperature by expanding to another decimal place (96 degrees, 96.44 degrees, 96.437 degrees, etc.).

Discrete and Continuous Random Variables

We can use a cumulative distribution function to calculate the probability of a specific range by taking the difference between two values from the cumulative distribution function. For example, to find the probability of observing between 3 and 6 heads, we can take the probability of observing 6 or fewer heads and subtracting the probability of observing 2 or fewer heads. This leaves a remnant of between 3 and 6 heads.

The visual to the right demonstrates how this works. It is important to note that to include the lower bound in the range, the value being subtracted should be one less than the lower bound. In this example, we wanted to know the probability from 3 to 6, which includes 3. Mathematically, this looks like the following equation:
```tex
\begin{aligned}
P(3<=X<=6) = P(X<=6) - P(X<3)\\
\text{or} \\
P(3<=X<=6) = P(X<=6) - P(X<=2)
\end{aligned}
```

Cumulative Distribution Function continued

A *random variable* is, in its simplest form, a function. In probability, we often use random variables to represent random events. For example, we could use a random variable to represent the outcome of a die roll: any number between one and six. 

Random variables must be numeric, meaning they always take on a number rather than a characteristic or quality. If we want to use a random variable to represent an event with non-numeric outcomes, we can choose numbers to represent those outcomes. For example, we could represent a coin flip as a random variable by assigning "heads" a value of 1 and "tails" a value of 0.

In this lesson, we will use `random.choice(a, size = size, replace = True/False)` from the `numpy` library to simulate random variables in python. In this method:

- `a` is a list or other object that has values we are sampling from
- `size` is a number that represents how many values to choose
- `replace` can be equal to `True` or `False`, and determines whether we keep a value in `a` after drawing it (`replace = True`) or remove it from the pool (`replace = False`). 


The following code simulates the outcome of rolling a fair die twice using `np.random.choice()`:
```python
import numpy as np

# 7 is not included in the range function
die_6 = range(1, 7)

rolls = np.random.choice(die_6, size = 2, replace = True)

print(rolls)
```
Output:
```
# [2, 5]
```

Random Variables

We can use the same `binom.pmf()` method from the `scipy.stats` library to calculate the probability of observing a range of values. As mentioned in a previous exercise, the `binom.pmf` method takes 3 values:
- `x`: the value of interest
- `n`: the sample size
- `p`: the probability of success

For example, we can calculate the probability of observing between 2 and 4 heads from 10 coin flips as follows:

```python
import scipy.stats as stats

# calculating P(2-4 heads) = P(2 heads) + P(3 heads) + P(4 heads) for flipping a coin 10 times
print(stats.binom.pmf(2, n=10, p=.5) + stats.binom.pmf(3, n=10, p=.5) + stats.binom.pmf(4, n=10, p=.5))
```
Output:
```
# 0.366211
```

We can also calculate the probability of observing less than a certain value, let's say 3 heads, by adding up the probabilities of the values below it:

```python
import scipy.stats as stats

# calculating P(less than 3 heads) = P(0 heads) + P(1 head) + P(2 heads) for flipping a coin 10 times
print(stats.binom.pmf(0, n=10, p=.5) + stats.binom.pmf(1, n=10, p=.5) + stats.binom.pmf(2, n=10, p=.5))
```
Output:
```
# 0.0546875
```

Note that because our desired range is less than 3 heads, we do not include that value in the summation.

When there are many possible values of interest, this task of adding up probabilities can be difficult. If we want to know the probability of observing 8 or fewer heads from 10 coin flips, we need to add up the values from 0 to 8:

```python
import scipy.stats as stats

stats.binom.pmf(0, n = 10, p = 0.5) + stats.binom.pmf(1, n = 10, p = 0.5) + stats.binom.pmf(2, n = 10, p = 0.5) + stats.binom.pmf(3, n = 10, p = 0.5) + stats.binom.pmf(4, n = 10, p = 0.5) + stats.binom.pmf(5, n = 10, p = 0.5) + stats.binom.pmf(6, n = 10, p = 0.5) + stats.binom.pmf(7, n = 10, p = 0.5) + stats.binom.pmf(8, n = 10, p = 0.5)
```
Output:
```
# 0.98926
```

This involves a lot of repetitive code. Instead, we can also use the fact that the sum of the probabilities for all possible values is equal to 1:

```tex
\begin{aligned}
P(0\;to\;8\;heads) + P(9\;to\;10\;heads) = P(0\;to\;10\;heads) = 1 \\
P(0\;to\;8\;heads) = 1 - P(9\;to\;10\;heads)
\end{aligned}
```

Now instead of summing up 9 values for the probabilities between 0 and 8 heads, we can do 1 minus the sum of two values and get the same result:

```python
import scipy.stats as stats
# less than or equal to 8
1 - (stats.binom.pmf(9, n=10, p=.5) + stats.binom.pmf(10, n=10, p=.5))
```
Output:
```
# 0.98926
```

Probability Mass Function Over a Range using Python

The _cumulative distribution function_ for a discrete random variable can be derived from the probability mass function. However, instead of the probability of observing a specific value, the cumulative distribution function gives the probability of observing a specific value OR LESS. 

As previously discussed, the probabilities for all possible values in a given probability distribution add up to 1. The value of a cumulative distribution function at a given value is equal to the sum of the probabilities lower than it, with a value of 1 for the largest possible number.

Cumulative distribution functions are constantly increasing, so for two different numbers that the random variable could take on, the value of the function will always be greater for the larger number. Mathematically, this is represented as:

```tex
\text{If}\; x_1 < x_2, \to CDF(x_1) < CDF(x_2)
```

We showed how the probability mass function can be used to calculate the probability of observing less than 3 heads out of 10 coin flips by adding up the probabilities of observing 0, 1, and 2 heads. The cumulative distribution function produces the same answer by evaluating the function at `CDF(X=2)`. In this case, using the CDF is simpler than the PMF because it requires one calculation rather than three.

The animation to the right shows the relationship between the probability mass function and the cumulative distribution function. The top plot is the PMF, while the bottom plot is the corresponding CDF. When looking at the graph of a CDF, each y-axis value is the sum of the probabilities less than or equal to it in the PMF.

Cumulative Distribution Function

Similar to how discrete random variables relate to probability mass functions, continuous random variables relate to probability density functions. They define the probability distributions of continuous random variables and span across all possible values that the given random variable can take on.

When graphed, a probability density function is a curve across all possible values the random variable can take on, and the total area under this curve adds up to 1.

The following image shows a probability density function. The highlighted area represents the probability of observing a value within the highlighted range.

![GIF or visual of the area under the curve highlighted and showing the calculated area under the curve](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/Adding-Area.gif)

In a probability density function, we cannot calculate the probability at a single point. This is because the area of the curve underneath a single point is always zero. The gif below showcases this.

![GIF or visual of the highlighted area under the curve getting smaller and smaller until the area equals 0](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/Normal-Distribution-Area-to-Zero.gif)

As we can see from the visual above, as the interval becomes smaller, the width of the area under the curve becomes smaller as well. When trying to evaluate the area under the curve at a specific point, the width of that area becomes 0, and therefore the probability equals 0.

We can calculate the area under the curve using the cumulative distribution function for the given probability distribution.

For example, heights fall under a type of probability distribution called a _normal distribution_. The parameters for the normal distribution are the mean and the standard deviation, and we use the form _Normal(mean, standard deviation)_ as shorthand. 

We know that women's heights have a mean of 167.64 cm with a standard deviation of 8 cm, which makes them fall under the Normal(167.64, 8) distribution. 

Let's say we want to know the probability that a randomly chosen woman is less than 158 cm tall. We can use the cumulative distribution function to calculate the area under the probability density function curve from 0 to 158 to find that probability.

![Image to show the area under the curve highlighted from 0 to 158 cm](https://static-assets.codecademy.com/skillpaths/master-stats-ii/probability-distributions/norm_pdf_167_8_filled.svg)

We can calculate the area of the blue region in Python using the `norm.cdf()` method from the `scipy.stats` library. This method takes on 3 values:
- `x`: the value of interest
- `loc`: the mean of the probability distribution
- `scale`: the standard deviation of the probability distribution


```python
import scipy.stats as stats

# stats.norm.cdf(x, loc, scale)
print(stats.norm.cdf(158, 167.64, 8))
```
Output:
```
# 0.1141
```

Probability Density Functions

The `binom.pmf()` method from the `scipy.stats` library can be used to calculate the PMF of the binomial distribution at any value. This method takes 3 values:

- `x`: the value of interest
- `n`: the number of trials
- `p`: the probability of success

For example, suppose we flip a fair coin 10 times and count the number of heads. We can use the `binom.pmf()` function to calculate the probability of observing 6 heads as follows:

```python
# import necessary library
import scipy.stats as stats

# st.binom.pmf(x, n, p)
print(stats.binom.pmf(6, 10, 0.5))
```
Output:
```
# 0.205078
```

Notice that two of the three values that go into the `stats.binomial.pmf()` method are the parameters that define the binomial distribution: `n` represents the number of trials and `p` represents the probability of success.