Question Video: Finding the Standard Deviation of a Discrete Random Variable | Nagwa Question Video: Finding the Standard Deviation of a Discrete Random Variable | Nagwa

Question Video: Finding the Standard Deviation of a Discrete Random Variable Mathematics • Third Year of Secondary School

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Let 𝑋 denote the discrete random variable that can take values 0, 2, 4, and 6. Given that 𝑃(𝑋 = 0) = 1/7, 𝑃(𝑋 = 2) = 2/7, and 𝑃(𝑋 = 4) = 2/7, find the standard deviation of 𝑋, giving your answer to two decimal places.

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Video Transcript

Let 𝑋 denote the discrete random variable that can take values zero, two, four, and six. Given that the probability 𝑋 equals zero is one-seventh, the probability 𝑋 equals two is two-sevenths, and the probability 𝑋 equals four is two-sevenths, find the standard deviation of 𝑋, giving your answer to two decimal places.

We’ve been told that this discrete random variable can take the values zero, two, four, and six. But we’ve only been given the probabilities for three of them. Before we can proceed with calculating the standard deviation, we need to find the probability that 𝑋 is equal to the fourth value, which is six. We can work this out because the sum of all the values in a probability distribution 𝑓 of 𝑋 must be equal to one. So one-seventh plus two-sevenths plus two-sevenths plus the probability 𝑋 is equal to six is equal to one. This simplifies to five-sevenths plus the probability 𝑋 equals six equals one. And then we can subtract five-sevenths from each side to find that the probability 𝑋 is equal to six is two-sevenths.

We may now find it helpful to express this probability distribution in a table, with the values in the range of the discrete random variable in the top row and their associated probabilities in the second row. So the probability distribution of 𝑋 can be represented as shown. We are asked to find the standard deviation of 𝑋, which is a measure of spread of its probability distribution. We use the Greek letter 𝜎, or sometimes 𝜎 subscript 𝑋, to denote the standard deviation. And it’s equal to the square root of the variance.

The variance of a discrete random variable 𝑋 can be calculated using the formula shown. It’s equal to the expected value of 𝑋 squared minus the expected value of 𝑋 squared. The difference in notation is really important here. In the second term, we find the expected or average value of 𝑋 and then we square it, whereas in the first term, we square the 𝑋-values first and then find their expected or average value. There’s quite a lot of work to be done here, so we’ll break it down into stages. And we’ll begin by calculating the expectation of 𝑋. This is the sum of each 𝑋-value in the probability distribution multiplied by its associated probability. And we can add a row to our table to work out these values.

First, we have zero multiplied by one-seventh, which is zero; then two multiplied by two-sevenths, which is four-sevenths; four multiplied by two-sevenths, which is eight-sevenths; and finally six multiplied by two-sevenths, which is twelve-sevenths. The expected value of 𝑋 is then the sum of these four values, which is 24 over seven. Next, we need to calculate the expected value of 𝑋 squared. The formula for this is the sum of each 𝑋 squared value multiplied by its corresponding probability. And the probabilities for 𝑋 squared are inherited directly from the probability distribution of 𝑋.

We can add another row to our table for the 𝑋 squared values, which are zero, four, 16, and 36. And then we’ll add another row in which we multiply these values by the probabilities. Zero multiplied by one-seventh is zero. Four multiplied by two-sevenths is eight-sevenths. 16 multiplied by two-sevenths is thirty-two sevenths. And finally, 36 multiplied by two-sevenths is seventy-two sevenths. The expectation of 𝑋 squared is the sum of these four values, which is 112 over seven.

So having calculated both the expectation of 𝑋 and the expectation of 𝑋 squared, we’re now able to calculate the variance of 𝑋. It’s equal to 112 over seven for the expectation of 𝑋 squared minus 24 over seven squared for the expectation of 𝑋 squared. And this is equal to 208 over 49. The final step in calculating the standard deviation is to take the square root of this value. 𝜎 then is the square root of 208 over 49, which in simplified surd form is four root 13 over seven. We can then evaluate this as a decimal, and it is 2.0603 continuing.

The question specifies that we should give our answer to two decimal places. So we round down to 2.06. The standard deviation of 𝑋 to two decimal places is 2.06, which means that, on average, observations of this discrete random variable 𝑋 are 2.06 units away from their mean.

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