Question

6-18.
Consider a random variable X with Mx = 40 and 0x = 5. Let Z=X+X+X+x.
a. Find uz and oz.
b. What does X+X+X+X represent?
What does 1/4 (X+X+X+X) represent?
d.Find mean 1/4z and variance 1/4z

Answer 1

In this problem, we consider the probability distribution of a random variable X with:

• mean value μ_X = 40,

,

• standard deviation σ_X = 5.

Now, we consider the random variable Z defined as:

$Z=X+X+X+X=4X\text{.}$

a. From probability theory, we know that a linear transformation of the type:

$X\rightarrow Z=a+b\cdot X,$

changes the mean value and the standard deviation in the following way:

$\begin{gathered} \mu_X\rightarrow\mu_Z=a+b\cdot\mu_X, \\ \sigma_X\rightarrow\sigma_Z=b\cdot\sigma_X\text{.} \end{gathered}$

In this case, we have a = 0 and b = 4, so we have that:

$\begin{gathered} \mu_X\rightarrow\mu_Z=0+4\cdot\mu_X=4\cdot40=160, \\ \sigma_X\rightarrow\sigma_Z=4\cdot\sigma_X=4\cdot5=20\text{.} \end{gathered}$

b. The sum Z = X + X + X + X = 4X represents a new random variable that is 4 times the original random variable X. So Z is a random variable obtained from a specific linear transformation of the original random variable X.

c. Because (X + X + X + X)/4 = X, this sum represents just the original random variable X.

d. To compute μ_Z/4 and σ_z/4, we take into account that:

$U=\frac{Z}{4}=\frac{(4X)}{4}=X\text{.}$

So the values of μ_Z/4 and σ_z/4 are just:

$\begin{gathered} \mu_{\frac{Z}{4}}=\mu_X=40, \\ \sigma_{\frac{Z}{4}}=\sigma_X=5. \end{gathered}$

Answers

a.

$\begin{gathered} \mu_Z=160, \\ \sigma_X=20. \end{gathered}$

b. The sum Z = X + X + X + X = 4X represents a new random variable that is 4 times the original random variable X. So Z is a random variable obtained from a specific linear transformation of the original random variable X.

c. Because (X + X + X + X)/4 = X, this sum represents just the original random variable X.

d.

$\begin{gathered} \mu_{\frac{Z}{4}}=\mu_X=40, \\ \sigma_{\frac{Z}{4}}=\sigma_X=5. \end{gathered}$