Final answer:
To compute the probabilities for the given standard normal random variable, we can use the cumulative distribution function (CDF) of the standard normal distribution. We can find probabilities such as P(Z > -1.0), P(Z > 1.5), and P(-3 < Z < 2.0) by either subtracting the CDF from 1 or by finding the difference between two CDFs. Using a standard normal probability table or a calculator, we can compute these probabilities.
Step-by-step explanation:
To compute the probabilities for the given standard normal random variable, we can use the cumulative distribution function (CDF) of the standard normal distribution. The CDF gives the probability that a standard normal random variable is less than or equal to a given value.
a. To find P(Z > -1.0), we can subtract the CDF at -1.0 from 1: P(Z > -1.0) = 1 - P(Z <= -1.0). Using a standard normal probability table or a calculator, we find P(Z <= -1.0) = 0.1587. Therefore, P(Z > -1.0) = 1 - 0.1587 = 0.8413.
b. To find P(Z > 1.5), we can use the same approach: P(Z > 1.5) = 1 - P(Z <= 1.5). Using the table or calculator, we find P(Z <= 1.5) = 0.9332. Therefore, P(Z > 1.5) = 1 - 0.9332 = 0.0668.
c. To find P(-3 < Z < 2.0), we need to find the probabilities for both -3 and 2.0 separately and then subtract them: P(-3 < Z < 2.0) = P(Z < 2.0) - P(Z < -3). Using the table or calculator, we find P(Z < 2.0) = 0.9772 and P(Z < -3) = 0.0013. Therefore, P(-3 < Z < 2.0) = 0.9772 - 0.0013 = 0.9759.