Question

X is a normal random variable with E[X] = -3 and V[X] = 4, compute a) ( ≤ 2.39) b) ( ≥ −2.39) c) (|| ≥ 2.39) d) (| + 3| ≥ 2.39) e) ( < 5) f) (|| < 5) g) With probability 0.33, variable X exceeds what value?

Answer 1

To compute the probabilities for different intervals and events using a normal random variable X, we can standardize the values and use the standard normal distribution table. The probabilities for the given intervals can be calculated by finding the corresponding z-scores. For the probability that X exceeds a certain value, we can use the inverse cumulative function to find the z-score and then convert it back to the original scale.

Step-by-step explanation:

To compute the given probabilities with a normal random variable X, we can use the standard normal distribution table. First, we need to standardize the values by subtracting the mean and dividing by the standard deviation. Then, we can look up the corresponding z-scores in the table to find the probabilities.

a) P(X ≤ 2.39) = P(Z ≤ (2.39 - (-3))/2) = P(Z ≤ 2.195) = 0.9857

b) P(X ≥ -2.39) = P(Z ≥ (-2.39 - (-3))/2) = P(Z ≥ 0.305) = 0.6184

c) P(|X| ≥ 2.39) = 2 * P(X ≥ 2.39) = 2 * (1 - P(X ≤ 2.39)) = 2 * (1 - 0.9857) = 0.0286

d) P(|X + 3| ≥ 2.39) = P(X + 3 ≥ 2.39) + P(-(X + 3) ≥ 2.39) = P(X ≥ -2.61) + P(X ≤ 5.39) = P(X ≥ -2.61) + (1 - P(X ≤ 5.39))

e) P(X < 5) = P(Z < (5 - (-3))/2) = P(Z < 4) = 1

f) P(|X| < 5) = 2 * P(X < 5) - 1 = 2 * 1 - 1 = 1

g) To find the value when X exceeds with probability 0.33, we can use the standard normal distribution inverse cumulative function (also known as the quantile function) to find the z-score corresponding to a cumulative probability of 0.67. From the table, the z-score is approximately 0.43. Then, we can convert the z-score back to the original scale by multiplying by the standard deviation and adding the mean: value = z * standard deviation + mean = 0.43 * 2 + (-3) = -2.14