FME2 Mindmap Week 9 2023

Probability Distributions：1. Ramdom Variables 2. Discrete Probability Distributions 3. Continuous Probability Distributions

Probability Distributions

• 1. Ramdom Variables
  • 1. A random variable (RV) is a numerical description of the outcome of an experiment. We use a capital X to denote random variables.
    • Random variables can be discrete or continuous.
  • 2. If random variable is discrete, this means the outcomes of the experiment can be measured in integers (whole numbers). This can be between an interval, or infinite.
  • 3. If a random variable is continuous, this means the outcomes of the experiment can take any value, it is not limited to integers. This can be between an interval, or a collection of intervals.
  • 4. Discrete RV's
    • 1. Expected value
      • 1. We can calculate some descriptive statistics for random variables.
      • 2. The mean, miu, of discrete random variables, is also called the Expected Value (E(X)). This is a measure of central tendency to see the average value of the RV.
      • 3. This is the sum of the individual possible random variable values, denoted by the small x, multiplied by the probability, p(x), that random variable will occur.
    • 2. Variance
      • 1. We also need a measure of dispersion/ variablity for our random variable, to see how spread out the RV values are. To do this we can calculate the Variance (Var(X)).
      • 2. This is the difference between the individual RV value and the mean, squared and then multiplied by the probability that RV values occur.
      • 3. We also commonly use the standard deviation to measure dispersion. This is the positive sqaure root of the variance.
    • 3. Example
    • 4. Linear combinations of Discrete RV's
      • 2. Sums of random variables
• 2. Discrete Probability Distributions
  • 1. The probability distribution for a random variable descreibes how probabilities are distributed over the values of the random variable.
  • 2. For discrete RV's, the probability distribution can be defined by a probability function, denoted by
  • 3. The probability function describes the probability for each possible vcalue of the random variable.
  • 4. Any probability function for a discrete random variable must satisfy two conditions
  • 5. The simplest example for a discrete RV is the Uniform Probability Function
    • n= the number of values the random variable may assume.
  • 6. Binomial probability distribution
    • 1. Properties
      • 1. The experiment consists of n identical trials.
      • 2. All the trials are independent.
      • 3. Each trial has two possible outcomes ,success or failure.
      • 4. The probability of success, represented by π is the same in every trial. Consequently so is the probability of failure (1-π).
    • 2. Conditional expectation
• 3. Continuous Probability Distributions
  • 1. Continuous random variables can take any value and are not limited to integers.
  • 2. Continuous probability distributions are probability distributions for continuous random variables.
  • 3. Define continuous probability distributions through a probability density function.
  • 4. The probability of the random variable occurring with an interval is the same as in any other equal sized interval.
  • 5. Uniform probability distribution-continuous
  • 6. Normal distribution
    • 1. The normal distribution is the most important probability distribution for continuous random variables.
    • 2. It has many practical applications including scientific measurements, test scores, amouts of rainfall, etc.
    • 3. Properties
      • 1. The normal distribution is characterized by a 'bell curve'.
      • 2. The highest point on the curve is the mean. It is also the mode and median. The mean can take any value - negative, zero or positive.
      • 3. The normal distribution is symmetric and the tails extend to infinity in both directions.
      • 4. The area to the left of the mean is equal to 0.5 and to the right, 0.5.
    • 4. Standard normal probability distribution
      • 1. With the normal distribution, we commonly denote the random variable with a Z instead of X.
      • 2. The standard normal distribution has a miu=0 and thita=1.
    • 5. Calculating probabilities
      • 1. The probability that the random variable Z will be less than or equal to a given value.
      • 2. The probability that the Z will take a value between two given values.
      • 3. The probability taht Z will be greater or equal to a given value.
    • 6. Statistic Tables
      • 4. We will start with the probability that Z<=a given value.
      • 1. The information in the table represents the area to the left of any Z value in a standard normal distribution.
    • 7. Other calculations
    • 8. Converting to the standard normal distribution
      • X= any normal random variable