Variance and Standard Deviation Christopher Croke University of Pennsylvania Math 115 Christopher Croke Calculus 115 This is a "weighted average". For a uniform distribution, i.e. 1 f (xi ) = N where N = #fxi g, then it is just the usual average: ΣN x E(X ) = i=1 i : N Problem: What is the expected number of heads in four tosses? 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 ): Christopher Croke Calculus 115 For a uniform distribution, i.e. 1 f (xi ) = N where N = #fxi g, then it is just the usual average: ΣN x E(X ) = i=1 i : N Problem: What is the expected number of heads in four tosses? 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". Christopher Croke Calculus 115 Problem: What is the expected number of heads in four tosses? 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. 1 f (xi ) = N where N = #fxi g, then it is just the usual average: ΣN x E(X ) = i=1 i : N 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. 1 f (xi ) = N where N = #fxi g, then it is just the usual average: ΣN x E(X ) = i=1 i : N Problem: What is the expected number of heads in four tosses? Christopher Croke Calculus 115 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 )? Expected value of a continuous random variable For a continuous random variable with probability density function f (x) on [A; B] we have: Z B E(X ) = xf (x)dx: A 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: Z B E(X ) = xf (x)dx: A 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 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: Z B E(X ) = xf (x)dx: A (note that this does not always exist if B = 1.) The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A Variance The first first important number describing a probability distribution is the mean or expected value E(X ). Christopher Croke Calculus 115 For continuous random variable X with probability density function f (x) defined on [A; B] we saw: Z B E(X ) = xf (x)dx: A (note that this does not always exist if B = 1.) The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A 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. Christopher Croke Calculus 115 (note that this does not always exist if B = 1.) The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A 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: Z B E(X ) = xf (x)dx: A Christopher Croke Calculus 115 The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A 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: Z B E(X ) = xf (x)dx: A (note that this does not always exist if B = 1.) 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: Z B E(X ) = xf (x)dx: A (note that this does not always exist if B = 1.) The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A 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: Z B E(X ) = xf (x)dx: A (note that this does not always exist if B = 1.) The variance will be: Z B σ2(X ) = Var(X ) = E((X − E(X ))2) = (x − E(X ))2f (x)dx: A Christopher Croke Calculus 115 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: 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. Christopher Croke Calculus 115 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: 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. Christopher Croke Calculus 115 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: 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? Christopher Croke Calculus 115 = E(X 2) + E(−2µX ) + E(µ2) = E(X 2) − 2µE(X ) + µ2 = E(X 2) − 2µ2 + µ2 = E(X 2) − µ2: 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) Christopher Croke Calculus 115 = E(X 2) − 2µE(X ) + µ2 = E(X 2) − 2µ2 + µ2 = E(X 2) − µ2: 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) 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 b) Find σ2. It is easier in this case to use the alternative definition of σ2: Z B σ2 = E(X 2) − E(X )2 = x2f (x)dx − E(X )2: A 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 Var(X )? Variance - Continuous case Problem Consider the problem of the age of a cell in a culture.