Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MATH 1070Q Section 6.4: The Normal Distribution Myron Minn-Thu-Aye University of Connecticut Objectives 1 Understand what a probability distribution is. 2 Understand the standard normal distribution and its table of probabilities. 3 Understand how to use the standard normal distribution to find out about any normal distribution. Probability distributions A probability distribution is a function associated to a random variable X that tells us the probability of X being within a certain range. The probability of X being between two values a and b is the area under the graph of the probability distribution over the interval on the x-axis from a to b: Normal distributions A normal distribution is a probability distribution with certain properties. Here are some examples: A normal distribution is . . . • Bell-shaped: values close to the mean are more likely, and the probabilities decrease as we move outwards. • Symmetric: values are equally likely to be above or below the mean. The standard normal distribution A random variable has a standard normal distribution if it is normally distributed (it has a graph similar to those on the previous slide) and: • the mean is µ = 0. • the standard deviation is σ = 1. A variable with a standard normal distribution is usually denoted Z (as opposed to X ). The standard normal distribution is so useful that we have a table listing many probabilities associated to it (p.390-391 in our text). The table for the standard normal distribution Look at the table of probabilities for the standard normal distribution. Let’s look up the probability associated to −1.13: look down the left column and find the row labeled by −1.1, then look across this row to the column labeled by 0.03. The probability in the table is 0.1292. This is telling us that P(Z ≤ −1.13) = 0.1292 We can see this graphically as: Calculations with the standard normal distribution Let Z be a random variable with a standard normal distribution. (a) Find P(Z ≤ 1.23) (b) Find P(Z ≥ 0.37) (c) Find P(−1.44 ≤ Z ≤ −0.12) Non-standard normal distributions What if X is a random variable with a normal distribution, but it’s not standard, meaning its mean is µ 6= 0 and its standard deviation is σ 6= 1? We can transform X into a standard normal variable using the formula: Z= X −µ σ For instance, suppose X is a normally distributed random variable with µ = 23 and σ = 4, and we want to know P(X ≤ 33). Heights of students Let the random variable X give the heights of students at the university. Suppose X is normally distributed with mean 68 inches and standard deviation 5 inches. (a) Find P(X ≤ 74) (b) Find P(X ≥ 77) Numbers of jelly beans A jelly bean company sells boxes where the number of jelly beans X in each box is normally distributed with mean 50 and standard deviation 6.25. (a) Find the probability that a box contains at least 55 jelly beans. (b) Find the probability that a box contains between 40 and 48 jelly beans. Recap 1 The area under the graph of a probability distribution tells us the probability of a random variable lying in a certain range of values. 2 A normal distribution has a bell-shaped curve. 3 We have a table of probabilities to tell us about the standard normal distribution, and we can transform any normal distribution into a standard normal distribution, then use probabilities from the table.