Define continuous random variable .

A continuous random variable is a random variable that can take any value within a range , rather than only certain separate values. Continuous random variables usually measure something, so height, weight and time are all continuous. That is the contrast with a discrete random variable, which counts, and the difference decides which distributions can model the variable.

What has to be true of a real-life variable before a normal distribution is a sensible model for it?

It must be continuous , so it measures something, its distribution must be symmetrical and bell-shaped with a single mode , and the population needs to be large enough. So, for example , a variable produced by a random number generator cannot be modelled this way, because every value is equally likely and it has no mode. Nor can how long a human lives, because that distribution is not symmetrical.

True or False? A normal distribution cannot model height, because it allows any real value and a height cannot be negative.

False. It is true that a normal distribution is defined for every real number, but values more than about four standard deviations from the mean have a probability density of practically zero. So a normal model of human height puts a negligible probability on the impossible values and describes the realistic ones well, which is what makes it usable for quantities like height and weight that have a natural floor. A model does not have to be perfect to be useful, only good enough over the range that matters.

A LevelMathsCambridge (CIE)Probability & Statistics 1FlashcardsStatistical DistributionsNormal Distribution

Normal Distribution (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Flashcards

Exam code: 9709

1/9

0Still learning

Know0

FrontThe Normal Distribution
Define continuous random variable.

Was this flashcard helpful?

Cards in this collection (9)

Define continuous random variable.
A continuous random variable is a random variable that can take any value within a range, rather than only certain separate values.
Continuous random variables usually measure something, so height, weight and time are all continuous.
That is the contrast with a discrete random variable, which counts, and the difference decides which distributions can model the variable.
For a continuous distribution, what is $P (X = k)$ , and what follows from it?
$P (X = k) = 0$ for every value of $k$ , because probability is the area under the graph and a single value is a line with no width.
What has a probability is a range of values: the area between $x = a$ and $x = b$ is $P (a \leq X \leq b)$ , and the total area under the graph is 1.
What follows is convenient: since the endpoints contribute nothing, strict and weak inequalities give the same answer.
$P (X \leq k) = P (X < k)$
True or False?
In $X ~ N (μ, σ^{2})$ , the second number in the bracket is the standard deviation.
False.
The second number is the variance, $σ^{2}$ : the notation says so, but it is easy to read past.
So $W ~ N (50, 36)$ has variance 36 and standard deviation $\sqrt{36} = 6$ .
This matters because calculators ask for the standard deviation, so entering the second number straight from the bracket is a common error; square root it first, unless the variance is already written as a square, as in $N (20, 5^{2})$ .
A normal distribution is symmetrical about $x = μ$ . What two things follow from that?
The three averages all coincide:
$mean = median = mode = μ$
and half the area lies on each side of the mean:
$P (X < μ) = P (X > μ) = 0.5$
The second is worth having ready. It gives you a probability with no calculation, and it is often the quickest way to check that an answer is on the right side of the mean.
Complete the proportions of a normal distribution that lie within one, two and three standard deviations of the mean:
within $μ \pm σ$ , about $______$ % of the data
within $μ \pm 2 σ$ , about $______$ % of the data
within $μ \pm 3 σ$ , about $______$ % of the data
The completed proportions are:
within $μ \pm σ$ , about 68% of the data, which is roughly two thirds
within $μ \pm 2 σ$ , about 95% of the data
within $μ \pm 3 σ$ , about 99.7% of the data, which is nearly all of it
These are worth knowing by heart as a sense check. If a calculation says that 40% of the data lies within one standard deviation of the mean, something has gone wrong.
Where are the points of inflection on a normal distribution curve?
At $x = μ \pm σ$ , exactly one standard deviation either side of the mean.
Those are the two places where the curve stops bending one way and starts bending the other, as it changes from falling ever more steeply to falling ever less steeply.
This gives you a way to read $σ$ off a sketch: it is the horizontal distance from the mean to a point of inflection.
How does a normal curve change when $μ$ changes, and how when $σ^{2}$ changes?
Changing $μ$ translates the curve horizontally, moving the whole shape along without altering it.
Changing $σ^{2}$ stretches it horizontally: a small variance gives a tall curve with a narrow centre, and a large variance a short curve with a wide centre.
The reason a narrower curve has to be taller is that the total area is always 1, so squeezing the curve inwards must push it upwards.
What has to be true of a real-life variable before a normal distribution is a sensible model for it?
It must be continuous, so it measures something, its distribution must be symmetrical and bell-shaped with a single mode, and the population needs to be large enough.
So, for example, a variable produced by a random number generator cannot be modelled this way, because every value is equally likely and it has no mode.
Nor can how long a human lives, because that distribution is not symmetrical.
True or False?
A normal distribution cannot model height, because it allows any real value and a height cannot be negative.
False.
It is true that a normal distribution is defined for every real number, but values more than about four standard deviations from the mean have a probability density of practically zero.
So a normal model of human height puts a negligible probability on the impossible values and describes the realistic ones well, which is what makes it usable for quantities like height and weight that have a natural floor.
A model does not have to be perfect to be useful, only good enough over the range that matters.

Sign up to unlock flashcards

By signing up you agree to our Terms and Privacy Policy

Previous:Basic Probability