Random Variables and Probability Distributions

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Random Variables and Probability Distributions
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Random Variables

• Random Variables are variables that may take on
  various values due to chance
• Random Variables are represented by probability
  distributions
• Probability distributions represent the values
  the RV can take on and the probability that the
  RV will take on those values
• Random Variables may be discrete or continuous

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Discrete Probability Distributions

• Discrete random variables may take on a countable
  many possible values
• A probability distribution for a discrete RV must
  represent the possible values for the RV and the
  probability that the RV takes on those values
• Probability distributions may be a table, graph
  or mathematical function

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Characteristics of Discrete Probability
Distributions

• The probability that the RV will take on any
  particular value must be between 0 and 1,
  inclusive.
• The sum of the probabilities for all possible
  values must be 1.

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Probability Distribution Example

• The roll of a fair die provides an example of a
  probability distribution table
• This is an example of a uniform distribution
• Other common discrete probability distributions
  are the binomial and Poisson

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Expected Value

• The expected value of a discrete random variable
  is the sum of the possible values weighted by
  their probabilities
• The expected value is not necessarily one of the
  possible values

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Variance of Discrete Random Variable

• The variance of a random variable describes the
  dispersion of the RV
• As usual, the standard deviation of the RV is the
  square root of the variance

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Covariance

• Covariance describes the way two random variables
  vary together
• It can tell you the nature of the relationship
  between the variables

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Portfolios

• Sometimes when investing you do not put all your
  eggs in one basket
• This is called diversifying
• A bundle of investments is called a portfolio
• Creating a portfolio can lower risk

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Portfolio Return

• You may want to know the expected return on your
  portfolio
• This expected return is a weighted average of the
  expected returns from the underlying investments

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Portfolio Risk

• You may also want to know your risk
• The portfolio risk is based on the risk of the
  underlying investments and their covariance
• The risk of the portfolio may be less than the
  risk of any of the underlying investments