Let z denote a random variable that has a standard normal distribution. Determine each of the fo

Let z denote a random variable that has a standard normal...

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is widely used in statistics because its properties, such as symmetry and the shape of its bell curve, allow for standardization of data and comparisons across different contexts.

Cumulative Distribution Function (CDF)

The cumulative distribution function, often denoted as ?(z) for the standard normal distribution, gives the probability that a random variable is less than or equal to a specified value. This function is essential for calculating probabilities and is typically referenced using standard normal tables.

Symmetry of the Normal Distribution

The symmetry of the standard normal distribution around the mean indicates that probabilities equidistant from the mean are equal. This property simplifies the computation of probabilities for events involving negative and positive values by reflecting one side to the other.

Complement Rule

The complement rule states that the probability of an event not occurring is equal to one minus the probability that it does occur. This rule is particularly useful when calculating tail probabilities, such as finding the probability of a value being greater than a given number.

Probability of Intervals

Calculating the probability of a random variable falling within a specific interval involves taking the difference between the cumulative probabilities at the endpoints of the interval. This method is fundamental when determining the probability for ranges of values in continuous distributions like the standard normal.

Transcript

00:01 Alright, so we're given a random variable probability distribution and we're asked to find the probability that the z -score is in one of these 8 situations.

00:12 So i'm not sure if you are required to find it using the calculator or the z -tables, but i'm going to do 4 with the z -tables and 4 with the calculator.

00:21 So the first one we're going to do z -tables.

00:23 So if you have a z -score, you know that your mean is 1 and your mean is 0 and your standard deviation is 1.

00:31 So that's your center line, that's your high point there.

00:38 And then you want to know the probability that a random z -score is less than 2 .36.

00:44 So 2 .36 is going to be somewhere up here.

00:46 So what you're looking for is this area, the area under the curve here, all of this.

00:52 So to do that, you're going to go to your z -table and you're going to go to the positive end and you're going to look up 2 .36.

01:00 So 2 .30123456 is right there, 9909.

01:11 And that tells you, that's the area under the curve.

01:15 So that's actually what we found here.

01:17 Our z is 2 .36.

01:19 We found this area to be 0 .9909.

01:24 So that's going to be this probability.

01:34 And when you're looking at a discrete random variable, it doesn't, the end point doesn't make any difference.

01:39 So this is just the probability that z is less than or equal to 2 .36.

01:44 It's going to be the same thing.

01:46 The end point is not going to make a difference because the numbers are so small to begin with.

01:54 So then the next one, the probability that your z -score is less than negative 0 .123.

01:59 So we want to do the same thing.

02:01 We want to get the area under the curve.

02:04 But in this case, we're looking at a z -score that's down here.

02:08 So we want this area.

02:11 So we're just going to use the negative z -score table.

02:14 So negative 1 .23 and 0 .2 right here is 0 .1093.

02:22 So that probability is 0 .1093.

02:29 And then the last one, these are a little bit trickier because we're looking at the probability that a z -score is between two scores.

02:38 So we want to know the probability that it falls somewhere between 1 .14 and 3 .35.

02:44 So what we're going to have to do here is we're going to find this area, the probability from here to here, and then we're going to do the same thing with the probability from here to here.

03:09 And what we're actually looking for is the space between.

03:14 So it's going to be the purple probability minus the green probability because what we're actually looking for is this right here.

03:24 So if you think about a number line, if i tell you that from here to here is 10 and from here to here is 4, then you know that this distance is going to be 6.

03:37 10 minus 4 is 6.

03:38 So that's what we're doing.

03:39 So we're going to look up 3 .35 first and i'm going to go to my table.

03:45 3 .35, 0, 1, 2, 3, 4, 5, 0 .9996.

03:55 And i'm going to subtract the probability that i get when i look up 1 .14.

04:00 So 1 .14, 0 .8729.

04:07 So that probability is 0 .1267.

04:15 Now for the next four, i'm going to do these on the calculator.

04:22 So if you've got a tia -4, then you're going to want to go to second distribution...