To find the probabilities and expected values using the standard normal table, we need to standardize the random variables by converting them to standard normal variables. We do this by subtracting the mean and dividing by the standard deviation.
1. P(X > -1):
First, we standardize -1 using the formula Z = (X - μ) / σ.
For X = -1, mean μ = 0 and standard deviation σ = 1.
Z = (-1 - 0) / 1 = -1.
Now, we find the probability using the standard normal table.
P(X > -1) = P(Z > -1).
From the standard normal table, the probability corresponding to -1 is 0.8413.
So, P(X > -1) = 1 - P(Z ≤ -1) = 1 - 0.8413 = 0.1587.
2. P(X ≤ -2):
We similarly standardize -2 using the formula Z = (X - μ) / σ.
For X = -2, mean μ = 0 and standard deviation σ = 1.
Z = (-2 - 0) / 1 = -2.
Now, we find the probability using the standard normal table.
P(X ≤ -2) = P(Z ≤ -2).
From the standard normal table, the probability corresponding to -2 is 0.0228.
So, P(X ≤ -2) = 0.0228.
3. Let V = (4 - Y) / 3.
To find the mean and variance of V, we need to calculate the mean and variance of Y.
Mean of Y (μ_y) = 3, Variance of Y (σ_y^2) = 16.
Now, substituting the formula V = (4 - Y) / 3, we get:
Mean of V (μ_v) = (4 - μ_y) / 3
= (4 - 3) / 3
= 1 / 3.
Variance of V (σ_v^2) = (1/3)^2 * σ_y^2
= (1/3)^2 * 16
= 16/9.
4. E[V] (Expected value of V) = Mean of V = 1/3.
5. Var(V) (Variance of V) = σ_v^2 = 16/9.
6. P(-2 < Y ≤ 2):
We first standardize -2 and 2 using the formula Z = (X - μ) / σ.
For Y = -2, mean μ = 3 and standard deviation σ = 4.
Z1 = (-2 - 3) / 4 = -5/4.
For Y = 2, mean μ = 3 and standard deviation σ = 4.
Z2 = (2 - 3) / 4 = -1/4.
Now, we find the probability using the standard normal table.
P(-2 < Y ≤ 2) = P(-5/4 < Z ≤ -1/4).
From the standard normal table, the probability corresponding to -5/4 is 0.1056,
and the probability corresponding to -1/4 is 0.4013.
So, P(-2 < Y ≤ 2) = 0.4013 - 0.1056 = 0.2957.
Remember to express your answers to an accuracy of 3 decimal places as mentioned in the question.