Question

Compute the following probabilities:

If Y is distributed N(−4,9),Pr(Y≤−5 )=
If Y is distributed N(−3,4),Pr(Y>−3) =
If Y is distributed N(−3,4),Pr(−6≤Y≤0) =

Answer 1

To compute the probabilities, convert the values to z-scores using the formula z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. For Y ~ N(-4, 9), Pr(Y ≤ -5) ≈ 0.1587. For Y ~ N(-3, 4), Pr(Y > -3) ≈ 0.5. For Y ~ N(-3, 4), Pr(-6 ≤ Y ≤ 0) ≈ 0.8664.

Step-by-step explanation:

To compute the probabilities, we will use the standard normal distribution table. Since Y is normally distributed, we need to convert the values to z-scores using the formula z = (x - μ) / σ, where μ is the mean and σ is the standard deviation.

1. For Y ~ N(-4, 9), Pr(Y ≤ -5) = Pr(z ≤ ( -5 - (-4) ) / sqrt(9)) = 0.1587
2. For Y ~ N(-3, 4), Pr(Y > -3) = Pr(z > ( -3 - (-3) ) / sqrt(4)) = 0.5
3. For Y ~ N(-3, 4), Pr(-6 ≤ Y ≤ 0) = Pr( ( -6 - (-3) ) / sqrt(4) ≤ z ≤ ( 0 - (-3) ) / sqrt(4) ) = Pr( -1.5 ≤ z ≤ 1.5 ) = Pr(z ≤ 1.5) - Pr(z ≤ -1.5) = 0.9332 - 0.0668 = 0.8664