Untitled Flashcards Set

Part 2

Discrete Random Variables

Random Variables refer to numerical values that describe the outcomes of a random process

• Discrete Random Variables takes a countable set of possible values with gaps between them on an number life 

• Continuous Random Variables is when it continues (such as distance).

Calculate Expected Value: Average result (expected result given probability) over many, many trails

Probability Distribution refers to the possible values of a random variable and their probabilities

• Is between 0 and 1, and has to add up to 1

• Can be deployed as histogram

  • Interpret: Shape of a probability, mean, and standard deviation

Mean/expected value/discrete random variable 

Its average value over many, many trials of the same random process.

• The mean, or expected value, is the long-run average value of the variable after many, many trials of the random process. It is denoted by 𝜇x or 𝐸 (𝑋).

• Finding SD: Measures how much the values of the random variable typically vary from the mean in many, many trials of the random process. 

Standard Deviation

Measure of the spread of a probability; it indicates how far, on average, data points deviate from the mean, with a larger standard deviation signifying a wider spread in the data

Variance

Standard deviation with squared differences; mathematically convenient 

Multiplying random variable by a constant

• Mean: multiples/divides by that constant

• Standard deviation:: multiples/devices 

• Variance: multiples/divides by that constant squared

• Shape: remains the same

Geometrics

Geometric distribution, are binary outcomes, independent trials, and the same probability of success for each trial, where the variable of interest is the number of trials needed to get the first success. 

• Example: Flipping a coin until heads appear is a geometric distribution. Each flip (trial) is independent, has a 50% chance of success (heads), and the variable of interest is the number of flips required to get the first head.

• Formula: P(X = x) = (1 - p)^(x-1) * p

"CDF" stands for "Cumulative Distribution Function," which means it calculates the probability that a random variable will be less than or equal to a specific value within a given distribution;, it accumulates the probabilities of all values below a certain point on the distribution curve

• Example: If you want to find the probability of getting at most 3 heads in 5 coin flips using a binomial distribution, you would use the "binomcdf" function on your calculator with the parameters (5, 0.5, 3).

“PDF" stands for “Probability Density Function” which represents a graph showing the probability of a continuous random variable taking on a specific value within a given range; essentially, it describes the "shape" of a distribution for continuous data.

Binomial Distributions

Binomial is a probability distribution that describes the likelihood of getting a specific number of "successes" in a fixed number of independent trials, where each trial has only two possible outcomes ("success" or "failure") and the probability of success remains constant throughout all trials

• It calculates the probability of getting a certain number of desired outcomes within a set number of attempts, with each attempt having the same chance of success.

Compute probabilities using the probability distribution of a discrete random variable

Calculate and interpret the mean (expected value) and standard deviation of a discrete random variable

• Show work by plugging into formulas (can be found on formula sheet)

• Know how to use your calculator to help you calculate

• Expected value is the average value from many, many trials. 

  • Standard deviation is the average deviation from the expected value from many, many trials. 

Compute probabilities using the probability distribution of a continuous random variable

• Distribution and parameters

• Boundary and direction

• Correct answer

Describe the effects of transforming a random variable by adding and/or multiplying by a constant

• Adding a constant → measures of center are affected

• Multiplying by a constant → measures of center and spread and affected

Find the mean and standard deviation of the sum or difference of independent random variables

• Mean, simply add or subtract

• Standard deviation, square root after you add the variances 

Find probabilities involving the sum and difference of independent Normal random variables

Determine whether conditions are met for using a binomial or geometric random variable

1. (B)inomial (success or failure)

2. (I)ndependence 

3. (N) is a set number (binomial) or until first success (geometric)

4. (S)ame probability of success

Calculate and interpret the mean and standard deviation of a binomial or geometric distribution 

• Use formula sheet to reference the mean and standard deviation of each 

Compute and interpret probabilities involving a binomial or geometric distribution

• Distribution and parameters 

  • Binomial, n=number of trials, p=probability

  • Geometric, p=probability

• Boundary and direction

• Correct answer

Determine when it is appropriate to use a Normal approximation to describe a binomial distribution to calculate probabilities 

• Check large counts condition

• np>10 and n(1-p)p>10