Final answer:
The student is inquiring about probabilities for a standard normal random variable X~N(0,1). To find the probabilities, one uses standard normal distribution tables or software. For non-standard normal variables, a transformation to standard normal is needed.
Step-by-step explanation:
The student is asking about the calculation of probabilities for a standard normal random variable (X), where X has a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1, denoted as X~ N(0,1). To answer such problems, one typically uses standard normal distribution tables or software to find the area under the curve between the given points, which represents the probability.
For example, finding the probability that X is between 1 and 4 would involve calculating the area under the standard normal distribution curve between x=1 and x=4. The calculation of these probabilities would depend on the z-scores representing these x-values. If the x-values were not already in the standard normal form, one would use the z-score formula: z = (x - μ) / σ.
Additionally, when the random variable is said to be a part of a different normal distribution (not standard), such as X~ N(1, 2) or X~ N(3, 5), a transformation to the standard normal distribution is needed before using the standard normal tables or software.