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  • Variance and Standard Deviation

    Christopher Croke

    University of Pennsylvania

    Math 115

    Christopher Croke Calculus 115

  • Expected value

    The expected value E (X ) of a discrete random variable X with probability function f (xi ) is:

    E (X ) = Σi xi · f (xi ).

    This is a ”weighted average”. For a uniform distribution, i.e. f (xi ) =

    1 N where N = #{xi}, then it is just the usual average:

    E (X ) = ΣNi=1xi N

    .

    Problem: What is the expected number of heads in four tosses?

    Christopher Croke Calculus 115

  • Expected value

    The expected value E (X ) of a discrete random variable X with probability function f (xi ) is:

    E (X ) = Σi xi · f (xi ).

    This is a ”weighted average”.

    For a uniform distribution, i.e. f (xi ) =

    1 N where N = #{xi}, then it is just the usual average:

    E (X ) = ΣNi=1xi N

    .

    Problem: What is the expected number of heads in four tosses?

    Christopher Croke Calculus 115

  • Expected value

    The expected value E (X ) of a discrete random variable X with probability function f (xi ) is:

    E (X ) = Σi xi · f (xi ).

    This is a ”weighted average”. For a uniform distribution, i.e. f (xi ) =

    1 N where N = #{xi}, then it is just the usual average:

    E (X ) = ΣNi=1xi N

    .

    Problem: What is the expected number of heads in four tosses?

    Christopher Croke Calculus 115

  • Expected value

    The expected value E (X ) of a discrete random variable X with probability function f (xi ) is:

    E (X ) = Σi xi · f (xi ).

    This is a ”weighted average”. For a uniform distribution, i.e. f (xi ) =

    1 N where N = #{xi}, then it is just the usual average:

    E (X ) = ΣNi=1xi N

    .

    Problem: What is the expected number of heads in four tosses?

    Christopher Croke Calculus 115

  • Expected value of a continuous random variable

    For a continuous random variable with probability density function f (x) on [A,B] we have:

    E (X ) =

    ∫ B A

    xf (x)dx .

    Problem: Consider our random variable X which is the sum of the coordinates of a point chosen randomly from [0, 1]× [0, 1]? What is E (X )?

    Christopher Croke Calculus 115

  • Expected value of a continuous random variable

    For a continuous random variable with probability density function f (x) on [A,B] we have:

    E (X ) =

    ∫ B A

    xf (x)dx .

    Problem: Consider our random variable X which is the sum of the coordinates of a point chosen randomly from [0, 1]× [0, 1]? What is E (X )?

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ).

    The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.) The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.) The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.) The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.)

    The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.) The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Variance

    The first first important number describing a probability distribution is the mean or expected value E (X ). The next one is the variance Var(X ) = σ2(X ). The square root of the variance σ is called the Standard Deviation.

    For continuous random variable X with probability density function f (x) defined on [A,B] we saw:

    E (X ) =

    ∫ B A

    xf (x)dx .

    (note that this does not always exist if B =∞.) The variance will be:

    σ2(X ) = Var(X ) = E ((X − E (X ))2) = ∫ B A

    (x − E (X ))2f (x)dx .

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2. Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2) = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2)− 2µ2 + µ2 = E (X 2)− µ2.

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2.

    Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2) = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2)− 2µ2 + µ2 = E (X 2)− µ2.

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2. Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2) = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2)− 2µ2 + µ2 = E (X 2)− µ2.

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2. Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2)

    = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2)− 2µ2 + µ2 = E (X 2)− µ2.

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2. Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2) = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2)− 2µ2 + µ2 = E (X 2)− µ2.

    Christopher Croke Calculus 115

  • Problem:(uniform probability on an interval) Let X be the random variable you get when you randomly choose a point in [0,B]. a) find the probability density function f . b) find the cumulative distribution function F (x). c) find E (X ) and Var(X ) = σ2.

    Alternative formula for variance: σ2 = E (X 2)− µ2. Why?

    E ((X − µ)2) = E (X 2 − 2µX + µ2) = E (X 2) + E (−2µX ) + E (µ2)

    = E (X 2)− 2µE (X ) + µ2 = E (X 2