Question

User
Let X be a random variable normally distributed with mean 1 and variance 2 . What is the probability that X belongs between 0.5 and 2?

Answer 1

To find the probability that X belongs between 0.5 and 2, we need to standardize the range of values using the z-score.

The formula for the z-score is:

z = (X - μ) / σ

Where:
X is the value of the random variable
μ is the mean
σ is the standard deviation (square root of the variance)

In this case, the mean (μ) is 1 and the standard deviation (σ) is the square root of 2, which is approximately 1.414.

Let's calculate the z-scores for 0.5 and 2:

z1 = (0.5 - 1) / 1.414 = -0.3536
z2 = (2 - 1) / 1.414 = 0.7071

Now, we need to find the probabilities associated with these z-scores using a standard normal distribution table or a calculator.

The probability that X belongs between 0.5 and 2 can be calculated as the difference between the cumulative probabilities at z2 and z1:

P(0.5 < X < 2) = P(0 < Z < 0.7071) - P(0 < Z < -0.3536)

Using a standard normal distribution table or calculator, we find:

P(0 < Z < 0.7071) ≈ 0.259
P(0 < Z < -0.3536) ≈ 0.363

Therefore, the probability that X belongs between 0.5 and 2 is approximately:

P(0.5 < X < 2) ≈ 0.259 - 0.363 ≈ -0.104.

Note that this probability cannot be negative, so there may be a mistake in the calculations or assumptions made. Please double-check the input values and the steps followed to verify the correct solution.