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Expectation & Variance of a Discrete Random Variable
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05
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We can get calculate the mean from the results of an experiment.
An equivalent number from a probability distribution is called the expectation or expected value. Expectation πΈ(π) Multiply each value of π by its corresponding probability. Add these together A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05 =βπ.π
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You may see the formula πΈ π =βπ₯π(π=π₯)
A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05 =βπ.π You may see the formula πΈ π =βπ₯π(π=π₯) Sum of
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You may see the formula πΈ π =βπ₯π(π=π₯)
A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05 =βπ.π You may see the formula πΈ π =βπ₯π(π=π₯) Sum of each π₯ times
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You may see the formula πΈ π =βπ₯π(π=π₯)
A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05 =βπ.π You may see the formula πΈ π =βπ₯π(π=π₯) Sum of each π₯ times Probability of π₯
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Variance You may see the formula πΈ π =βπ₯π(π=π₯)
A d.r.v. π has a probability distribution as shown. Find πΈ(π) π₯ β2 β1 1 2 π(π=π₯) 0.3 0.1 0.15 0.4 0.05 πΈ π = β2 Γ0.3+ β1 Γ0.1+0Γ0.15+1Γ0.4+2Γ0.05 =βπ.π You may see the formula πΈ π =βπ₯π(π=π₯) Sum of each π₯ times Probability of π₯ Variance
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How did we find variance of a set of data previously in S1?
Variance of discrete random variables How did we find variance of a set of data previously in S1?
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Our earlier slide Variance and Standard Deviation
These are measures of spread around the mean. They use all data values in their calculations. There are two formulas for finding the Variance (π 2 ) β both always give the same result. π 2 = Ξ£ π₯β π₯ 2 π π 2 = Ξ£ π₯ 2 π β Ξ£π₯ π 2 βThe mean of the squares minus the square of the meanβ You need to memorise both formulas. In almost every situation the second formula is easier to use.
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Variance of discrete random variables
We do essentially the same thing, though in S1 we use πππ for a d.r.v πππ(π)=πΈ π 2 β πΈ π 2 The following will show how to calculate πΈ π 2 as part of the calculation of πππ How did we find variance of a set of data previously in S1? The mean of the squares minus the square of the mean π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π)
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π π π(π=π₯) 0.4 0.2 0.3 0.1
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 π(π=π₯) 0.4 0.2 0.3 0.1
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 π(π=π₯) 0.4 0.2 0.3 0.1
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 π(π=π₯) 0.4 0.2 0.3 0.1
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1
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Variance of discrete random variables
π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1 a) πΈ π = 1Γ Γ Γ0.3 +(4Γ0.1) =2.1
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For πΈ( π 2 ) we just use the π₯ 2 row instead
Variance of discrete random variables π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1 a) πΈ π = 1Γ Γ Γ0.3 +(4Γ0.1) =2.1 For πΈ( π 2 ) we just use the π₯ 2 row instead
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For πΈ( π 2 ) we just use the π₯ 2 row instead
Variance of discrete random variables π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1 a) πΈ π = 1Γ Γ Γ0.3 +(4Γ0.1) =2.1 For πΈ( π 2 ) we just use the π₯ 2 row instead
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For πΈ( π 2 ) we just use the π₯ 2 row instead
Variance of discrete random variables π₯ 1 2 3 4 π(π=π₯) 0.4 0.2 0.3 0.1 For the given distribution, find a) πΈ π b) πΈ π 2 c) πππ(π) We need to include an π₯ 2 row π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1 a) πΈ π = 1Γ Γ Γ0.3 +(4Γ0.1) =2.1 For πΈ( π 2 ) we just use the π₯ 2 row instead b) πΈ π 2 = 1Γ Γ Γ0.3 +(16Γ0.1) =5.5
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Variance of discrete random variables
π₯ 1 2 3 4 π₯ 2 9 16 π(π=π₯) 0.4 0.2 0.3 0.1 a) πΈ π = 1Γ Γ Γ0.3 +(4Γ0.1) =2.1 b) πΈ π 2 = 1Γ Γ Γ0.3 +(16Γ0.1) =5.5 πππ(π)=πΈ π 2 β πΈ π 2 Ex 8C page 162 Q1-5 (about 10 β 15 mins) Ex 8D page 164 Skip Q3 c) πππ π =5.5 β =1.09
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Hints 8C 8D Ex 8C page 162 Q1-5 (about 10 β 15 mins) Ex 8D page 164
Q3c β simply show what each is & therefore show whether equal or not Q4 β List the sample space (all the equally possible outcomes), starting HH,β¦ Q5 β make two equations with π and π in each & solve simultaneously 8D Q1a β you should be able to work it out in your head! Q1b β show working for this part Q4a β Use the sample space diagram at right (draw your own in an exam if necessary) Q5 β Draw your own sample space Q6 β List sample space Q7 β Think about the expected value, by looking at the probabilities of the higher and lower valuesβ¦ Remember βWrite downβ means no working required. Ex 8C page 162 Q1-5 (about 10 β 15 mins) Ex 8D page 164 Skip Q3
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