# Stats - Statistical Distributions

> Source: https://ollybritton.com/notes/a-level/maths/topics/stats/statistical-distributions/ · Updated: 2021-01-26 · Tags: stats, probability, distributions, year-1, school

## See Also
- [Stats - Binomial Distribution](https://ollybritton.com/notes/a-level/maths/topics/stats/binomial-distribution/)
- [Stats - Probability](https://ollybritton.com/notes/a-level/maths/topics/stats/probability/)

## Flashcards
##### What is a random variable??
A variable whose value depends on the outcome of a random event.

##### What is the notation for a random variable??
Capital letters, $X$.

##### What is the notation for the random variable $X$ taking a particular value $x$??
$$
P(X = x)
$$

##### Can you explain $P(X = x)
The probability that the outcome of the random variable $X$ was the specific outcome $x$.

##### What's the shorthand for $P(X = heads)$??
$$
p(heads)
$$

##### What is a discrete uniform distribution??
A distribution that can only take specific values, which all have the same probability for each outcome.

##### What is a piecewise function??
We choose the function from a list depending on the input.

##### ![PHOTO PROBABILITY FUNCTION](probability-function.png) What is $P(X = 2)$??
$$
0.2
$$

##### ![PHOTO PROBABILITY TABLE](probability-table.png) What is $P(X = 3)$??
$$
0.3
$$

##### What is the advantage of writing probability distributions as a function??
You can have a rule or expression based on the outcome.

##### What is the advantageo f writing probability distributions as a table??
It's easier to see the probability for each outcome.

##### What is the underlying sample space for $X$, the number of heads when three coins are tossed??
$$
HHH, HHT, HTT, HTH, THH, THT, TTH, TTT
$$

##### What was the head of the table be for the probability distribution of $X$, the number of heads when three coins are tossed??
Number of heads, $x$.

##### In a probability distribution table where all possible outcomes are listed, what must the sum of all probabilities equal??
$$
1
$$

##### What would that notation for "the sum of all probabilities"??
$$
\sum p(x)
$$

##### ![PHOTO PROBABILITY TABLE RANGE](probability-table-range.png) What is $P(X > 3)$??
$$
0.6
$$

##### For a discrete random variable, what is $p(x)$ called??
The probability mass function.

##### Why is $p(x)$ called a probability mass function for a discrete random variable??
Because the probability of each outcome represents an actual amount of the overall probability.

##### Visualise what the graphical probability distribution would look like for a six-sided die??
![PHOTO GRAPHICAL PROBABILITY DISTRIBUTION](graphical-probability-distribution.png)

### 2022-06-02
##### "Helen believes that the random variable C, representing cloud cover from the large data set, can be modelled by a discrete uniform distribution" How could you write down the probability distribution??
Use the fact that cloud cover can only be one of 9 possible values, so the probability of each case is $\frac{1}{9}$.

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