03/24/2020

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- The following topics will be covered in this lecture:
- Uniform distribution
- Normal distributions
- Finding the area corresponding to a probability
- Finding a score corresponding to an area
- Critical values
- General normal distributions

Courtesy of Mario Triola, *Essentials of Statistics*, 6th edition

- So far our examples have focused on
**discrete random variables**, e.g.: - Results of
**coin flips**– \( x \) is modeled with a**binomial distribution**. - Results of
**success / failure trials**– \( x \) is modeled with a**binomial distribution**. - Number of
**occurances in an interval**– \( x \) is modeled with a**Poisson distribution**. - We will now turn our attention to
**continuous random variables**, but we will use what we learned about**discrete variables to motivate this**. - Recall that the probability histogram had the property, \[ \begin{align} \text{Area of Rectangle }x_\alpha &= P(x=x_\alpha) \times 1\\ &= P(x=x_\alpha). \end{align} \]

- We also saw that we have the property \[ \sum_{x_\alpha \in \mathbf{R}} P(x=x_\alpha) =1. \]
- Putting the above two properties together, we know, \[ \sum_{x_\alpha \in \mathbf{R}} \text{Area of Rectangle }x_\alpha =1. \]
- For
**continuous random variables**, we in fact have the**same property with a minor modification**:Let \( f(x) \) describe a curve for a probability distribution. Then the total area under the curve \( f(x) \) equals \( 1 \), and the probability of any event \( A \) equals the associated area under \( f(x) \) for all \( x_\alpha \) in the case of \( A \).

Courtesy of Ania Panorska CC

- A basic example of the area property is with the
**uniform distribution** - Let’s suppose that we are studying some procedure where all outcomes are equally likely.
- A very simple example is if you are asked to
**guess a random number**between \( 1 \) and \( 10 \),**but including decimals**. - That is, we will suppose that guessing \( 1.23453453 \) is equally likely as guessing \( 5 \).

- Viewed in the above,
- Our
**random experiment**is**guessing some number**. - The
**outcome**is**one guess**. - The
**random variable \( x \)**is**assigned the value of the guess**.

- Our
- Because we allow
**arbitrary decimal expansions**, there are**infinitely many choices**. - However,
**all choices lie in the finite range \( [1,10] \) and are equally likely**. **Discuss with a neighbor:**if the area under the curve \( f(x) \) for \( x_\alpha \) in \( [1, 10] \) must equal one, and the height of \( f(x) \) is constant, what is the height?