Suppose X is a random variable with mean µX and standard deviation σX. Its z-score is the random variable. z= X- µX/ σXRequried:What is the mean, µz, and standard deviation, σz, of Z? Begin by re-writing Z so that it is in the form Z = a + bX. What are a and b in this case?

Mean of the random variable z = μz = 0

Standard deviation of the random variable z = σz = 1

From the z-score, z = (X - μₓ)/σₓ

Comparing with Z = a + bX

a = (-μₓ/σₓ)

b = (1/σₓ)

Step-by-step explanation:

Complete Question

Suppose X is a random variable with mean μₓ and standard deviation σₓ. Its z-score is the random variable, z = (X - μₓ)/σₓ

Required: What is the mean, µz, and standard deviation, σz, of Z?

Begin by re-writing Z so that it is in the form Z = a + bX. What are a and b in this case?

Solution

X is a random variable with mean μₓ and standard deviation σₓ,

The z-score = z = (X - μₓ)/σₓ

If Z is a random variable too, what is the mean and standard deviation of random variable Z.

z = (X - μₓ)/σₓ

Zσₓ = X - μₓ

Z = (-μₓ/σₓ) + (1/σₓ)X

Z = a + bX

a = (-μₓ/σₓ)

b = (1/σₓ)

To obtain the mean of random variable Z

At the mean point, X = μₓ

Z = (-μₓ/σₓ) + (1/σₓ)X

Becomes

μz = (-μₓ/σₓ) + (1/σₓ)μₓ

μz = (-μₓ/σₓ) + (μₓ/σₓ) = 0

For the standard deviation, for one standard deviation away from the mean, X - μₓ = σₓ

So,

Z = (X - μₓ)/σₓ

At one standard deviation away from the mean

σz = (X - μₓ)/σₓ

And X - μₓ = σₓ

Hence,

σz = (σₓ/σₓ) = 1

Hope this Helps!!!

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