WEBVTT
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The function in the given table is a probability function of a discrete random variable π.
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Given that the expected value of π is 6.5, find the standard deviation of π.
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Give your answer to two decimal places.
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So weβve been given a table containing the possible values of π₯ and the probabilities of these possible values.
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However, one of the possible values is missing.
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And instead, we have π΄.
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So we need to find the value of π΄.
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In the question, weβre given that the expected value is 6.5.
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And we can use this to help us find the value of π΄.
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Letβs recall the equation for the expected value.
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The expected value or πΈ of π is equal to the sum of π₯π times π of π₯π.
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And what this means is we take each of the possible values of π₯ and multiply them by their given probabilities and then add all of these products together.
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So letβs calculate what πΈ of π is.
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πΈ of π is equal to three times 0.2 plus π΄ times 0.1 plus six times 0.1 plus eight times 0.6.
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Now we can simplify this.
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And we get πΈ of π is equal to 0.6 plus 0.1π΄ plus 0.6 plus 4.8.
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And this is equivalent to six plus 0.1π΄.
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Now from the question, we have that the expected value is equal to 6.5.
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And so, therefore, we have two different values for πΈ of π.
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And we can put them equivalent to one another.
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And so we get that 6.5 is equal to six plus 0.1π΄.
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Now we can subtract six from either side.
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And this gives us 0.5 is equal to 0.1π΄.
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Now itβs a little tricky to divide by 0.1.
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So letβs write 0.1 as a fraction.
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And we get that 0.5 is equal to one tenth times π΄.
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Now we simply multiply both sides by 10.
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π΄ is equal to five.
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Now letβs redraw our table with the value of π΄ equals five.
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We are now ready to calculate the standard deviation.
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Letβs recall the equation for calculating the standard deviation.
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We have that the standard deviation is equal to the square root of the expected value of the squares minus the square of the expected value.
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We will calculate each of these components individually.
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So starting with the expected value of the squares, we need to remember the equation for this.
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So πΈ of π squared is equal to the sum from one to four of π₯π squared timesed by π of π₯π.
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So what that means is we take the square of each of our possible values and then multiply them by their respective probabilities and add them all together.
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πΈ of π squared is equal to three squared times 0.2 plus five squared times 0.1 plus six squared times 0.1 plus eight squared times 0.6.
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Now letβs expand the squares.
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And this gives us nine times 0.2 plus 25 times 0.1 plus 36 times 0.1 plus 64 times 0.6.
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Next, weβll multiply through.
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And we get 1.8 plus 2.5 plus 3.6 plus 38.4.
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Then adding this all together gives us that the expected value of the squares is equal to 46.3.
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Now letβs calculate the square of the expected value.
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And now we can given πΈ of π in the question.
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So this is equal to 6.5 squared, which is just 42.25.
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Now we have all the values we need to calculate our standard deviation.
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So letβs substitute these into the equation.
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This gives us that the standard deviation is equal to the square root of 46.3 minus 42.5.
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Now we just type this into our calculator, remembering to round our answer to two decimal places.
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This gives us a solution that the standard deviation is equal to 2.01.