Reading Quiz 7

Question 1

Random variable

Answer 1

A variable whose value is a numerical outcome of a random phenomenon

Question 2

Discrete random variable

Answer 2

Random variable which takes on only a finite or countable number of possible values

Question 3

Probability distribution of a random variable X

Answer 3

Tells the possible values of X and tells how probabilities are assigned to those values
Also called a probability function

Question 4

Continuous random variable

Answer 4

Takes all values in an interval of numbers

Question 5

Probability distribution of X

Answer 5

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

Question 6

Probability density curve

Answer 6

Aka density curve

Continuous random variable’s probability distribution is described by this graph

Question 7

Two important characteristics of density curves

Answer 7

1. The area under the graph of a density curve equals 1, corresponding to total probability 1
2. The graph of a density curve lies on or above the horizontal axis

Question 8

Continuous probability for individual outcomes

Answer 8

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

Question 9

How to picture a probability distribution

Answer 9

Drawing a probability histogram in the discrete case

Graphing the density curve in the continuous case

Question 10

How to identify random variable of interest

Answer 10

X = number of ——— for discrete random variables
X = amount of —–—— for continuous random variables

Question 11

mean of any discrete random variable

Answer 11

aka expected value
weighted average of the possible outcomes
each outcome weighted by probability
multiply value by probability than add

Question 12

variance of a discrete random variable

Answer 12

value minus mean quantity squared then times probability, add all

Question 13

how to find mean and variance on calc

Answer 13

stats calc 1 (L1, L2)

Question 14

standard deviation

Answer 14

o with line on top

square root of variance

Question 15

law of large numbers

Answer 15

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

Question 16

rules for means

Answer 16

1. if X is random variable and a and b are fixed numbers then μ(a+bX) = a + bμX
2. if X and Y are random variables then μ(X+Y) = μX + μY
   and μ(X-Y) = μX - μY

Question 17

rules for variances

Answer 17

1. if X is a random variable and a and b are fixed numbers then δsquared(a+bX) = (b^2) (δ^2(X))
2. 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

Question 18

addition rule for variances

Answer 18

implies that standard deviations don’t generally add

standard deviations most easily combined by using rules for variances

Question 19

extra

Answer 19

any linear combination of independent normal random variables is also normally distributed

Question 20

A random variable is a variable whose value is a ________ of a random phenomenon.

Answer 20

numerical outcome

Question 21

A random variable with a countable number of possible values is a _____ random variable.

Answer 21

discrete

Question 22

What is a probability distribution of a discrete random variable?

Answer 22

A. A list of the values the variable can take on, and the probability for each value.

Question 23

For the probability distribution of a discrete random variable, every probability is between ___ and ___, and the sum of all the probabilities is equal to ___.

Answer 23

0 and 1, 1

Question 24

In a probability histogram, what quantity do the horizontal and vertical axes represent?

Answer 24

A. The horizontal axis represents the possible values the random variable can take on, and the vertical axis represents the probability of that value.

Question 25

A continuous random variable can take on how many values for a certain interval in its domain?

Answer 25

an infinite number

Question 26

A continuous random variable’s probability distribution is described by a graph called the ___.

Answer 26

A. Density curve, or probability density curve. (This is the graph of the probability density function, or pdf.)

Question 27

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?

Answer 27

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.

Question 28

Suppose you have a continuous random variable X. What is the probability that X=10?

Answer 28

A. Zero. Continuous probability distributions assign probability 0 to every individual outcome.

Question 29

In a continuous probability distribution, what is the relationship between the probability that X<10 and the probability that X  10?

Answer 29

A. The two are equal, because the probability that X=10 is 0.

Question 30

True or false: the normal distribution is an example of a continuous probability distribution.

Answer 30

true

Question 31

The mean of a discrete random variable is the sum of the products of all the possible values and the ______.

Answer 31

probabilities of those values

Question 32

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?

Answer 32

1

Question 33

The mean of a random variable is often called the e_____ v_____ of the variable.

Answer 33

expected value

Question 34

The mean of symmetric continuous probability distributions lies at the ____ of the curves.

Answer 34

center

Question 35

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.

Answer 35

probability

Question 36

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.)

Answer 36

A. (0-1)2*1/3 + (1-1)2*1/3 + (2-1)2*1/3 = 2/3.

Question 37

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?

Answer 37

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.

Question 38

19. What is the law of large numbers, in your own words?

Answer 38

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.

Question 39

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%.

Answer 39

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.

Question 40

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.

Answer 40

true

Question 41

The mean of the sum of two random variables equals what?

Answer 41

A. The sum of the means of the two variables.

Question 42

A linear combination of two independent normally distributed random variables is distributed how?

Answer 42

normally