Reading Quiz 7 Flashcards
Random variable
A variable whose value is a numerical outcome of a random phenomenon
Discrete random variable
Random variable which takes on only a finite or countable number of possible values
Probability distribution of a random variable X
Tells the possible values of X and tells how probabilities are assigned to those values
Also called a probability function
Continuous random variable
Takes all values in an interval of numbers
Probability distribution of X
Described by a density curve
Probability of any event is the area under the density curve and above the values of X that make up the event
Probability density curve
Aka density curve
Continuous random variable’s probability distribution is described by this graph
Two important characteristics of density curves
- The area under the graph of a density curve equals 1, corresponding to total probability 1
- The graph of a density curve lies on or above the horizontal axis
Continuous probability for individual outcomes
0 because there is no area above a point
Thus can ignore distinction between > and _> when finding probabilities for continuous (but NOT DISCRETE) random variables
How to picture a probability distribution
Drawing a probability histogram in the discrete case
Graphing the density curve in the continuous case
How to identify random variable of interest
X = number of ——— for discrete random variables X = amount of —–—— for continuous random variables
mean of any discrete random variable
aka expected value
weighted average of the possible outcomes
each outcome weighted by probability
multiply value by probability than add
variance of a discrete random variable
value minus mean quantity squared then times probability, add all
how to find mean and variance on calc
stats calc 1 (L1, L2)
standard deviation
o with line on top
square root of variance
law of large numbers
difference between a sample proportion or mean and a population proportion or mean gets smaller as sample size gets larger
average of values of X observed in many trials must approach μ
rules for means
- if X is random variable and a and b are fixed numbers then μ(a+bX) = a + bμX
- if X and Y are random variables then μ(X+Y) = μX + μY
and μ(X-Y) = μX - μY
rules for variances
- if X is a random variable and a and b are fixed numbers then δsquared(a+bX) = (b^2) (δ^2(X))
- if X and Y are independent (IMPORTANT) random variables then δ^2(X+Y) = δ^2(X) + δ^2(Y) and δ^2(X-Y) = δ^2(X) + δ^2(Y)
called addition rule for variances of independent random variables
addition rule for variances
implies that standard deviations don’t generally add
standard deviations most easily combined by using rules for variances
extra
any linear combination of independent normal random variables is also normally distributed
A random variable is a variable whose value is a ________ of a random phenomenon.
numerical outcome
A random variable with a countable number of possible values is a _____ random variable.
discrete
What is a probability distribution of a discrete random variable?
A. A list of the values the variable can take on, and the probability for each value.
For the probability distribution of a discrete random variable, every probability is between ___ and ___, and the sum of all the probabilities is equal to ___.
0 and 1, 1
In a probability histogram, what quantity do the horizontal and vertical axes represent?
A. The horizontal axis represents the possible values the random variable can take on, and the vertical axis represents the probability of that value.
A continuous random variable can take on how many values for a certain interval in its domain?
an infinite number
A continuous random variable’s probability distribution is described by a graph called the ___.
A. Density curve, or probability density curve. (This is the graph of the probability density function, or
pdf.)
Events, for continuous random variables, are described by the random variable’s taking on a value within a certain interval. The probability of that event is represented by what aspect of the density curve?
A. The area under the curve, between the two points that bound the interval, or the area under the curve, over the values (on the x-axis) that make up the event.
Suppose you have a continuous random variable X. What is the probability that X=10?
A. Zero. Continuous probability distributions assign probability 0 to every individual outcome.
In a continuous probability distribution, what is the relationship between the probability that X<10 and the probability that X 10?
A. The two are equal, because the probability that X=10 is 0.
True or false: the normal distribution is an example of a continuous probability distribution.
true
The mean of a discrete random variable is the sum of the products of all the possible values and the ______.
probabilities of those values
Suppose there are two possible outcomes for a certain random variable, 0 and 100. The probability of getting 0 is .99 and the probability of getting 100 is .01. What is the mean of the random variable?
1
The mean of a random variable is often called the e_____ v_____ of the variable.
expected value
The mean of symmetric continuous probability distributions lies at the ____ of the curves.
center
The variance of a discrete random variable is the sum of the products of the squared deviation of each possible value from the mean of the distribution and the _____ for that value.
probability
Suppose there is a distribution with possible values 0, 1, and 2, each with probability 1/3. What is the variance, i.e. the sigma-squared, of this distribution? (This is also known as the variance of the population.)
A. (0-1)21/3 + (1-1)21/3 + (2-1)2*1/3 = 2/3.
Think back to the definition of the variance of a sample. Suppose you had a sample consisting of 0, 1, and 2, with mean 1. Is the variance of this sample the same as the variance of the population?
A. No. The variance of the sample is the sum of the squared deviations over n-1. So the variance of the sample would be (1+0+1)/(3-1) or 1, rather than 2/3.
- What is the law of large numbers, in your own words?
A. One way of putting it is that as the sample size approaches infinity, the sample mean approaches the population mean. Another is that you can make the sample mean get as close as you want to the population mean by getting a large enough sample.
True or false: If by chance, you flip a coin and get 10 heads in a row, the law of large numbers tells us that if we flip many more times, we will get just a tiny bit under 50% heads in the remaining tosses, to compensate for the first 10 heads and make the long-range probability equal 50%.
A. False. The definition of independent trials implies that the coin “doesn’t remember” the first 10 flips and the subsequent results are not influenced by the initial ones.
True or false: the mean of a linear function of a random variable is that same linear function of the mean of the random variable. In other words, the mean of a + bX is a +b*the mean of X.
true
The mean of the sum of two random variables equals what?
A. The sum of the means of the two variables.
A linear combination of two independent normally distributed random variables is distributed how?
normally
