Question

Let X be a Gaussian random variable with μ=0 and σ=2.

a) Find out P[X>2]
b) Find out P[X<−2]
c) Find out P[−2≤X≤2]
d) Find out P[X=0]
e) Find out P[X=2]

Answer 1

Let X follow a Gaussian distribution with a mean of 0 and a standard deviation of 2. To find probabilities for specific intervals or points, areas under the curve of the normal distribution are calculated using the cumulative distribution function. The probability of a continuous random variable taking a specific value is always 0.

Step-by-step explanation:

Let X be a Gaussian random variable specified by the question. This means X follows a normal distribution with mean (μ) of 0 and a standard deviation (σ) of 2. To solve the provided problems, we use the properties of the normal distribution.

1. To find P[X>2], we calculate the area under the normal curve to the right of X=2. This is equivalent to 1 minus the cumulative distribution function (CDF) at X=2.
2. For P[X<-2], we calculate the area under the normal curve to the left of X=-2. This can be directly found using the CDF at X=-2.
3. P[-2≤X≤2] can be found by calculating the area within the bounds of X=-2 and X=2, which is the CDF at X=2 minus the CDF at X=-2.
4. The probability P[X=0] in a continuous distribution like the normal distribution is always 0 because the probability of a continuous random variable taking on any exact value is zero.
5. Similarly, P[X=2] is also 0 for the same reason mentioned above.

For proper calculations, we would typically use statistical tables or software to find the CDF values for the specified points, which then allow us to determine the probabilities.