Final answer:
In this question, we are asked to find probabilities for a standard normal random variable. We calculate the probabilities using the cumulative distribution function (CDF) of the standard normal distribution.
Step-by-step explanation:
In this question, we are given a standard normal random variable, which is denoted by Z for simplicity. The standard normal distribution has a mean of 0 and a standard deviation of 1.
a. To find the probability that Z is less than 0, we use the cumulative distribution function (CDF) of the standard normal distribution. P(Z < 0) is equal to the area under the curve to the left of 0, which is 0.5.
b. To find P(|Z| > 2), we can translate this into finding P(Z < -2 or Z > 2). We can calculate this by subtracting P(Z < -2) from 1 and then doubling the result. Since the standard normal distribution is symmetric, P(Z < -2) is the same as P(Z > 2), so we can calculate P(|Z| > 2) as 1 - 2 * P(Z < -2), which is approximately 0.0456.
c. To find P(|Z| < 1), we can calculate P(Z < -1 or Z > 1) and subtract it from 1. This is the same as finding 1 - P(|Z| > 1). Using the result from part b, P(|Z| > 1) is approximately 0.3173, so P(|Z| < 1) is approximately 1 - 0.3173, which is approximately 0.6827.