Question

A Gaussian random variable, X has mean 3 and variance 9 , which we often write as X∼N(3,9). (a) Find P(X≥12), the probability that X is greater than or equal to 12 , in terms of the Q(⋅) function.

Answer 1

To find P(X≥12) for X~N(3,9), calculate the z-score (which is 3 in this case) and express the probability as the Q function of the z-score, Q(3), representing the tail probability to the right of z=3 on the standard normal curve.

Step-by-step explanation:

The question requires finding the probability that a Gaussian random variable X, which is normally distributed with mean 3 and variance 9 (X~N(3,9)), is greater than or equal to 12. To find this probability P(X≥12), we use the Q function, which is defined as the tail probability of the standard normal distribution. First, we standardize the variable by calculating the z-score.

The z-score is given by:

z = (X - μ) / σ

For X=12, μ=3 (mean), σ=√9=3 (standard deviation), the z-score is:

z = (12 - 3) / 3 = 3

Now we can express the probability using the Q function:

P(X≥12) = Q(z) = Q(3)

Since the Q function is the tail probability, P(X≥12) represents the area under the standard normal curve to the right of the z-score of 3.