Given a standard normal distribution, find the area under the curve that lies (a) to the left of

Given a standard normal distribution, find the area under...

Key Concepts

Standard Normal Distribution

The standard normal distribution is a specific type of normal distribution with a mean of zero and a standard deviation of one. It serves as a reference distribution in statistics for understanding probabilities, percentiles, and the behavior of normally distributed data. Its bell-shaped curve is symmetric, which simplifies the calculation of areas and tail probabilities when determining likelihoods associated with specific z-scores.

Z-Score

A z-score represents the number of standard deviations a particular value is away from the mean of a distribution. In a standard normal distribution, z-scores are used to relate raw scores to the standardized scale, which then allows the computation of corresponding probabilities or percentiles using statistical tables or software tools.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a normal distribution indicates the probability that a random variable will take a value less than or equal to a specific threshold. It is essential for determining areas under the curve, such as the probability to the left of a given z-score or between two z-scores.

Tail and Interval Probabilities

Tail probabilities refer to the likelihood that a value falls in the extreme ends (either the left or right tail) of the distribution's curve, while interval probabilities represent the area under the curve between two specified z-scores. Understanding how to compute these probabilities using symmetry and properties of the cumulative distribution function is crucial for solving a wide range of statistical problems involving the normal distribution.

Transcript

00:01 This problem explores the normal distribution and specifically the standard normal distribution, and we're going to look at finding some probabilities under the curve of the standard normal distribution.

00:14 So to start with, the f of x given here is shown here in the top left is actually the equation of the standard normal distribution.

00:24 And the standard normal distribution means that we have a mu of zero, mu equal to zero, and we have a standard deviation of 1.

00:33 This is just by definition that is what the standard normal distribution parameter value is equal to.

00:42 And the z score is given by this function here.

00:48 It's x minus mu over sigma, but in this case again we know that mu is zero and we know that sigma is one.

00:57 So the z score is actually the x value directly only in the case of the standard normal distribution.

01:04 Now let's look at a value here.

01:10 Some random example.

01:17 Let's see.

01:21 So now let's say that we wanted to find the area under the curve to the left of a z score of negative 1.

01:29 So i've drawn a very rough standard normal distribution here on the right and the z score is given underneath.

01:38 Usually the z score is here and it's the values of the z score card.

01:43 To where you are on the curve.

01:45 So it's centered at zero, which means it's a right in the middle, which means that 50 % of the curve is on the left and 50 % of the curve is on the right.

01:55 So, and it goes increments this way in positives, two, three, and one, two, three.

02:02 It's just mirrored.

02:03 It's just the same thing the other way around.

02:05 So if we wanted to look at, let's say, area under the curve, area under curve where we have a z score equal 10 minus 1 and since we know as before the z score equaling minus 1 is means that the x actually is equal to minus 1 directly as well so now that so what we're interested is in is on the left side right on the left side so let's draw in an example here this is the reason that we are interested in.

02:52 How do we get that region? now there's actually two ways to do this, and i'm going to show you the first way and the second way.

03:01 The first would be to normally, as usual, when we're finding probabilities, we want to integrate something.

03:07 Now this curve actually extends out to all the way to minus infinity.

03:11 It goes forever.

03:12 And on the same on the right side, it goes to positive infinity.

03:16 So in this case, our lower bound would be minus infinity.

03:21 And we know that our z score, or and our x value here is going to be minus 1.

03:28 And we are want to integrate our distribution, which is given here, and this is the equation.

03:34 So we know that this is a constant, so we're going to pull that out.

03:37 So this is going to be 1 over square root 2, pi here...

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