Kanya Shah

Jed Quiaoit

It's very useful to **transform** a random variable by adding or subtracting a constant or multiplying or dividing by a constant. This can help to simplify calculations or to make the results easier to interpret. 🪄

This section is about transforming random variables by adding/subtracting or multiplying/dividing by a constant. At the end of this section, you'll know how to combine random variables to calculate and interpret the mean and standard deviation.

When you transform a random variable by adding or subtracting a constant, the mean and standard deviation of the transformed variable are also shifted by the same constant. For example, if you have a random variable, X, with mean E(X) and standard deviation SD(X), and you transform it to a new random variable, Y, by adding a constant, c, to each value of X, then the mean and standard deviation of Y are given by: ➕➖

Similarly, if you transform a random variable by multiplying or dividing it by a constant, the mean and standard deviation of the transformed variable are also multiplied or divided by the same constant. For example, if you have a random variable, X, with mean E(X) and standard deviation SD(X), and you transform it to a new random variable, Y, by multiplying each value of X by a constant, c, then the mean and standard deviation of Y are given by: ✖️➗

When you transform a random variable by adding or subtracting a constant, it affects the **measures of center and location**, but it does not affect the **variability** or the shape of the distribution.

When you transform a random variable by multiplying or dividing it by a constant, it affects the **measures of center, location**, and **variability**, but it does not change the shape of the distribution.

It is also possible to combine two or more random variables to create a new random variable. To calculate the mean and standard deviation of the combined random variable, you would need to use the formulas for the expected value and standard deviation of a linear combination of random variables.

For example, if you have two random variables, X and Y, with means E(X) and E(Y) and standard deviations SD(X) and SD(Y), respectively, and you want to create a new random variable, Z, by adding them together, then the mean of Z is given by:

💡 **Summary**

For example, if you have two random variables, X and Y, with means E(X) and E(Y) and standard deviations SD(X) and SD(Y), respectively, and you want to create a new random variable, Z, by adding them together, then the standard deviation of Z is given by:

💡 **Summary**

🎥 **Watch: AP Stats - ****Combining Random Variables**

Try this one on your own! 😉

Two random variables, X and Y, represent the number of hours a student spends studying for a math test and the number of hours a student spends studying for a science test, respectively. The probability distributions of X and Y are shown in the tables below:

A new random variable, Z, represents the total number of hours a student spends studying for both tests.

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