Probability Distributions Random variable A random variable

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Probability Distributions
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Random variable

• A random variable is a variable that represents a
  number in a experiment.
• Example In having 3 children, we can get 0, 1,
  2, or 3 girls. The number of girls is a random
  variable.

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Discrete or Continuous

• Probability distributions can be discrete or
  continuous.
• Discrete comes from a counting process. This
  lesson and the binomial lesson are for discrete
• Continuous comes from a measuring process. The
  normal curve is continuous.

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Probability distribution

• A probability distribution is a list of all the
  values of the random variable and their
  corresponding probabilities.
• To be valid, each probability must be between 0
  and 1 AND the sum of the probabilities must equal
  1.

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Example 1

• Valid?

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Example 2

• Valid?

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Example 3

• Valid?

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Example 4 Girls in a family of 3

• This chart is for the number of girls in a family
  of 3 children (from the tree diagram done in the
  previous lesson)
• Valid?

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Graph

• The graph is a histogram. The bars do not need
  to be connected as the data is discrete.

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Mean of a probability distribution

• The mean of a probability distribution is
  calculated by
• Each value of x is multiplied by its
  corresponding probability. All of these answers
  are then added.

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Mean for the number of girls
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Standard deviation

• The formula is
• Steps
• 1. Subtract m from each x
• 2. Square the answer
• 3. Multiply by the probability
• 4. Add values
• 5. Obtain the square root.

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Calculating
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• Report to one more decimal place than the values
  for x.

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Using the TI to do the calculations

• Put the x values and the probability values into
  2 lists.

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• At the top of list 3, type L1 L2

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• Press enter

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• Quit the lists
• Press 2nd, List, Math, sum

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• Specify the list

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• Store this value in A, by pressing STO, followed
  by A.

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• At the top of List 4, type (L1-A)2L2

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• Press enter

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• Quit the list, obtain the sum.
• Take the square root.

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Your turn
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Unusual events Girls example

• It is unusual to have values of x beyond 2
  standard deviations of the mean
• Are there any unusual events?

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Expected value

• The mean is also sometimes called the expected
  value.
• We can think of it as the average value if we
  were to continue the trials indefinitely.

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Example

• When you give a casino 5 for betting on the
  number 7 in roulette, there is a 1/38 probability
  that you will win 175. You have a 37/38
  probability of losing your 5.

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Calculate the expected value
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Another example

• A life insurance policy costs 350. There is a
  0.999 probability that the person will live that
  year. If the person dies, the insurance company
  pays 300,000. Find the expected revenue, per
  policy, for the insurance company.

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Calculate the expected value