Let Z ~N(0, 1). Find P(-2.1 < Z < -1.1).
Answer (2)
The probability that a standard normal random variable Z lies between -2.1 and -1.1 is approximately 0.1178. This was calculated using the CDF functions for the standard normal distribution. By applying the property of symmetry and computing necessary CDF values, we arrive at the final answer.
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Express the probability using the CDF: P ( − 2.1 < Z < − 1.1 ) = Φ ( − 1.1 ) − Φ ( − 2.1 ) .
Use the symmetry property: Φ ( − x ) = 1 − Φ ( x ) , so P ( − 2.1 < Z < − 1.1 ) = Φ ( 2.1 ) − Φ ( 1.1 ) .
Find the CDF values: Φ ( 2.1 ) ≈ 0.9821 and Φ ( 1.1 ) ≈ 0.8643 .
Calculate the probability: P ( − 2.1 < Z < − 1.1 ) ≈ 0.9821 − 0.8643 = 0.1178 .
Explanation
Understand the problem and provided data We are given a standard normal random variable Z, which follows a normal distribution with a mean of 0 and a standard deviation of 1, denoted as Z ~ N(0, 1). Our goal is to find the probability that Z lies between -2.1 and -1.1, i.e., P(-2.1 < Z < -1.1).
Express the probability using CDF To find the probability P(-2.1 < Z < -1.1), we can use the cumulative distribution function (CDF) of the standard normal distribution, denoted by Φ(z). The CDF gives the probability that a standard normal random variable is less than or equal to z. We can express the desired probability as the difference between the CDF values at the upper and lower bounds: P(-2.1 < Z < -1.1) = Φ(-1.1) - Φ(-2.1).
Use symmetry property of normal distribution Since the standard normal distribution is symmetric around 0, we can use the property that Φ(-x) = 1 - Φ(x) to rewrite the CDF values for negative z-scores in terms of positive z-scores: Φ(-1.1) = 1 - Φ(1.1) and Φ(-2.1) = 1 - Φ(2.1). Substituting these expressions into our probability equation, we get: P(-2.1 < Z < -1.1) = (1 - Φ(1.1)) - (1 - Φ(2.1)) = Φ(2.1) - Φ(1.1).
Find CDF values Now we need to find the values of Φ(2.1) and Φ(1.1). Using a standard normal distribution table or a calculator with statistical functions, we find that Φ(2.1) ≈ 0.9821 and Φ(1.1) ≈ 0.8643.
Calculate the probability Finally, we calculate the difference to find the desired probability: P(-2.1 < Z < -1.1) = Φ(2.1) - Φ(1.1) ≈ 0.9821 - 0.8643 = 0.1178.
State the final answer Therefore, the probability that Z lies between -2.1 and -1.1 is approximately 0.1178.
Examples
Understanding probabilities related to the standard normal distribution is crucial in many fields. For instance, in finance, it helps in assessing the risk associated with investments. If you're analyzing stock returns and find they are normally distributed, you can use these probabilities to estimate the likelihood of returns falling within a certain range, helping investors make informed decisions. For example, calculating the probability of a stock's return being between -2.1 and -1.1 standard deviations from the mean can provide insights into potential losses.
