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