Suppose that X is a random variable with mean and variance . Define a new variable Y such that . Which of the following is a true statement about the probability distribution of Y?
There are no statements provided in the question.
The given equation defines a new random variable Y as the difference between the random variable X and its mean μ, divided by the square root of its variance σ^2.
Y = (X - μ) / √(σ^2)
The true statement about the probability distribution of Y is that Y follows a standard normal distribution, denoted as N(0, 1). This means that Y has a mean of 0 and a variance of 1.
To determine the probability distribution of Y, we need to understand how Y is related to X. The given definition of Y tells us that Y is obtained by subtracting the mean of X, μ, from X, and then dividing the result by the square root of the variance of X, σ.
Thus, we can express Y mathematically as:
Y = (X - μ) / √σ
Now, let's analyze the possible statements about the probability distribution of Y to identify which one is true.
Statement 1: The mean of Y is zero.
To check this statement, we need to calculate the mean of Y. Since we know the mean of X is μ, distributing the terms in the equation gives:
E(Y) = E((X - μ) / √σ)
= (1 / √σ) * E(X - μ)
= (1 / √σ) * (E(X) - E(μ))
Since E(μ) = μ, we can simplify this to:
E(Y) = (1 / √σ) * (μ - μ)
= (1 / √σ) * 0
= 0
Therefore, Statement 1 is true - the mean of Y is indeed zero.
Statement 2: The variance of Y is one.
To check this statement, we need to calculate the variance of Y. Using the properties of variances, we have:
Var(Y) = Var((X - μ) / √σ)
= (1 / σ) * Var(X - μ)
= (1 / σ) * Var(X)
Since the variance of X is σ, we can simplify this to:
Var(Y) = (1 / σ) * σ
= 1
Therefore, Statement 2 is true - the variance of Y is indeed one.
In conclusion, both Statement 1 and Statement 2 are true about the probability distribution of Y.