Ch. 7: Continuous Random Variables Flashcards
A _________ random variable may assume any numerical value in one or more intervals.
continuous
(ex: car mileage, temperature, etc.)
T or F: You should use a continuous probability distribution to assign probabilities to intervals of values.
True
(Will use continuous when we have continuous data like height, weight, time, distance, interest rates, etc.; stuff we can measure)
(Most likely these examples ^ have units)
The ________ is the continuous probability distribution of the continuous random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval.
curve f(x)
(know this definition; write on cheat sheet; look in camera roll for visual example of this definition on page titled Continuous Probability Distributions)
Other names for a continuous probability distribution are ________ _______ and ________ _______ _______.
- probability curve
- probability density function
What are the 3 types of continuous probability distributions?
- Uniform
- Normal
- Exponential
What are the Properties of Continuous Probability Distributions?
Properties of f(x): f(x) is a continuous function such that:
1.) f(x) ≥ 0 for all x
2.) The total area under the curve of f(x) is equal to 1
Essential point: An area under a continuous probability distribution is a probability
(look at camera roll and write these on cheat sheet)
What is the shape of the curves for the following distributions:
1. Uniform Distribution
2. Normal Distribution
3. Exponential Distribution
- Rectangle
- Bell-shaped
- Decreasing smooth
(look at camera roll for pics of these)
Under continuous random variables, finding the probability just means finding the…
area under the curve
T or F: For continuous random variables, you should always pay attention to the signs it is asking, but not as important for discrete.
True
(ex: pay attention to the signs =, ≥, or ≤ when asked to find the probability)
(ex: find the probability when x≤c or something like that)
T or F: For continuous random variables, you can’t find the probability of a single point.
True
(ex: if asked to find the probability when x=c, and c is just a line on the graph, then the probability would be 0)
Continuous Random Variables:
- If asked to find P(x≤a), you should find the area on the (left/right) side of A under the curve.
- If ask to find P(x≥a), you should find the area on the (left/right) side of A under the curve.
- If asked to find P(a≤x≤b), this means we are trying to find the _____ under this curve.
- left-side (same thing as P(x<a)
- right-side (same thing as P(x>a)
- area (finding probability of x between a and b)
Uniform Distribution is also called a _________ distribution, and is a probability that has a __________ probability.
rectangular; constant
KNOW THIS
What is the formula for finding the area of a rectangle?
When should you use this formula?
- Area of rectangle = length x height
- For uniform distribution
For uniform distribution, you are asked to find P(c≤x≤d). The formula for finding the area under a rectangle is length x height.
How do you calculate the length? The height?
- Length = d - c
- Height = 1/(d-c)
(This height is only good when our x is less than d or greater than c. Anything greater than d or less than c, height is going to be zero)
(remember– area under the curve is = to 1)
(look at picture titled Uniform Distribution in camera roll for further explanation; 2 pics)
For Uniform Distribution, how do you calculate the following:
1. Mean?
2. Variance?
3. Standard Deviation?
- Mean = (c+d) / 2
- Variance = (d-c)^2 / 12
- Standard Deviation = (d-c) / square root of 12
(look in camera roll for these formulas; write on cheat sheet)
Look at example A in camera roll for example on how to do Uniform Distribution.
Ok (3 pics total)
Look at example 7.6 in camera roll for more practice on how to solve probabilities for uniform distribution.
Ok (3 pics total)
With a uniform distribution problem, if asked to find P({0 ≤ x ≤ 2} or {5 ≤ x ≤ 6}), what do you do once you find both of these probabilities for each interval?
Add the two probabilities together.
(or means ADD)
T or F: For Normal Probability Distribution, the normal curve is symmetrical around M (mean) and the total area under the curve equals 1.
True
Normal distribution curve is what shape?
Bell-shaped
What are the 5 properties of the Normal Distribution?
- There are an INFINITE number of normal curves.
- The highest point is over the mean (which is also the median and mode)
- The curve is symmetrical about its mean (the left and right halves of the curve are mirror images of each other)
- The tails of the normal extend to infinity in both directions (the tails gets closer to the horizontal axis but never touch it)
- The area under the normal curve to the right of the mean equals the area under the normal curve to the left of the mean. (the area under each half is 0.5)
(KNOW THESE PROPERTIES! Write on cheat sheet; in camera roll)
The shape and position of any individual normal curve depends on its specific _______ and _______ ______.
mean; standard deviation
KNOW THIS
For Normal Distribution, how does the following affect the curve?
1.) Mean
a.) Small mean
b.) Large mean
2.) Small standard deviation
3.) Large standard deviation
1.) moves the curve left or right on number line.
a.) small mean = moves curve left
b.) large mean = moves curve right
2.) our data will be around the mean, so the curve is narrow and peaked
3.) our data is spread out from the mean, so the curve is wide/flat/spread out.
T or F: For normal distribution, a higher standard deviation is more risky, while a lower standard deviation is less risky.
True
