Let $Z \sim N(0,1)$. Find $a$ so that $P(Z < a) = 0.90$.
$a=$
Answer (2)
The value of a such that P ( Z < a ) = 0.90 for a standard normal distribution is approximately 1.2816. This represents the 90th percentile of the standard normal distribution, where 90% of values lie below this z-score.
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We are given a standard normal distribution Z ∼ N ( 0 , 1 ) and want to find a such that P ( Z < a ) = 0.56 .
We need to find the inverse of the standard normal CDF, Φ − 1 ( 0.56 ) .
Using a calculator or standard normal table, we find a ≈ 0.1509692154967774 .
Therefore, the value of a is 0.1509692154967774 .
Explanation
Understand the problem and provided data We are given that Z follows a standard normal distribution, denoted as Z ∼ N ( 0 , 1 ) . This means that Z has a mean of 0 and a standard deviation of 1. We want to find the value a such that the probability that Z is less than a is equal to 0.56, i.e., P ( Z < a ) = 0.56 . In other words, we are looking for the 56th percentile of the standard normal distribution.
Finding the inverse CDF To find the value of a , we need to use the inverse of the standard normal cumulative distribution function (CDF), also known as the quantile function or probit function. We want to find a such that Φ ( a ) = 0.56 , where Φ is the standard normal CDF. This is equivalent to finding a = Φ − 1 ( 0.56 ) .
Calculate the value of a Using a calculator or a standard normal table, we can find the value of a that corresponds to a probability of 0.56. The result of this operation is approximately a = 0.1509692154967774 .
Final Answer Therefore, the value of a such that P ( Z < a ) = 0.56 is approximately 0.151.
Examples
In finance, the standard normal distribution is used to model stock prices and other financial variables. If you want to find the stock price such that there is a 56% chance that the actual stock price will be below that value, you would use the inverse CDF of the standard normal distribution, similar to this problem. For example, if stock price changes follow a standard normal distribution, finding the value 'a' such that P(Z < a) = 0.56 helps determine a threshold below which the stock price is expected to fall 56% of the time. This is useful for risk management and setting stop-loss orders.
