L4 Continuous random variables

Continuous random variables

Any values in a paricular interval could be an outcome, either one or multiple.

E.g. weight of a person.

Characteristics: Continuous random variables

1. Infinite Values: Since there are infinitely many possible values, it is not possible to determine the probability that a continuous random variable assumes a single specified value (P(X=x)=0).

2. Interval Focus: We must instead calculate the probability that the variable falls within a specified interval (e.g., P(85≤X≤90)).

3. Density Function: Instead of relying on P(X=x), continuous variables use the probability density function . The expected value is calculated using an integral: E(X)=∫xf(x)

Probability density function f(x)

A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.

Why use PDF?

- Tells how likely the values are distributed across an interval

- tells how dense (tät) the probability is arount that value.

Normal Distribution (the Gaussian distribution)

Means that observations are symmetrically distributed around the mean value. Most values closer to mean and fewer further outside. Forms a bellshaped curve.

Normal distribution characteristics

- Descriptive model defined as a continuous frequency distribution of infinite range.

- A variable X that is normally distributed with mean μ and variance σ2 is written as X∼N(μ,σ2).

• Properties: The curve is bell-shaped and symmetric around the mean (μ). The mean and median are equal, and the total area under the curve is 1 (100%)

The Standard Normal Distribution (Z)

Normal distribution where the variable Z has a mean (μ) equal to 0 and a variance (σ2) equal to 1. It is written as Z∼N(0,1).

• Key Z-Values: Important Z-values are used for confidence intervals and hypothesis testing (e.g., P(−1.960≤Z≤1.960)=0.95)

Standardization Theorem

Allows us to convert any normal random variable X into the standard normal variable Z.

• Formula: Z=σX−μ​.

• This conversion enables us to use the standard normal tables to find the probability for X

The Inverse Transformation

This method is used when we know the probability (or Z-score) and need to find the corresponding value of X.

• Formula: x=μ+zσ.

• This process allows us to set values (like a guarantee length) based on a required probability (e.g., finding the X value such that P(X≤x)=0.1)