6.2 - Mean and SD of Combinations and Transformations

Front

What is the mean of a random variable after adding a constant?

Back

If a random variable X has a mean μ and a constant c is added to it, the new mean Y is given by: Y = μ + c.

Front

How does adding a constant affect the standard deviation of a random variable?

Back

Adding a constant to a random variable does not change its standard deviation. If X has a standard deviation σ, then Y = X + c has the same standard deviation σ.

Front

What is the formula to convert feet to inches?

Back

To convert feet to inches, multiply the number of feet by 12. (inches = feet × 12)

Front

What is the mean of a random variable in a different unit?

Back

To find the mean in a different unit, convert the original mean using the conversion factor. For example, if the mean height is 5.8 ft, in inches it is 5.8 × 12.

Front

What is the standard deviation of a random variable when converting units?

Back

To convert the standard deviation from one unit to another, multiply by the conversion factor. For example, if the standard deviation is 0.24 ft, in inches it is 0.24 × 12.

Front

What is the mean of a random variable after a linear transformation?

Back

For a linear transformation Y = aX + b, the mean of Y is given by: E[Y] = aE[X] + b.

Front

What is the standard deviation of a random variable after a linear transformation?

Back

For a linear transformation Y = aX + b, the standard deviation of Y is given by: SD[Y] = |a| SD[X].

Front

How do you find the mean of a sum of random variables?

Back

The mean of the sum of two random variables X and Y is the sum of their means: E[X + Y] = E[X] + E[Y].

Front

What is the standard deviation of the sum of independent random variables?

Back

For independent random variables X and Y, the standard deviation of their sum is: SD[X + Y] = sqrt(SD[X]^2 + SD[Y]^2).

Front

What is the effect of adding a constant to a random variable on its distribution?

Back

Adding a constant shifts the distribution of the random variable to the right (if positive) or left (if negative) without changing its shape.