AP Stat Ch 6

Question 0

Discrete random variable

Answer 0

A random variable is discrete if its set of possible values is a collection of isolated points on the number line. A discrete random variable has a countable number of possible values.

Question 1

Random variable

Answer 1

A numerical variable whose value depends on the outcome of a chance experience.

Question 2

Properties of a discrete random variable, X

Answer 2

0<=P(X)<=1

Sum of P(X)=1

Question 3

Continuous random variable

Answer 3

A random variable is continuous if it’s set of possible values includes an entire interval on the number line. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.

Question 4

Properties of density functions

Answer 4

1. f(x)>= 0
2. Total area under curve = 1
3. Probability that x falls in any particular interval is e area under curve in that interval

Question 5

How to calculate z score

Answer 5

Z = X- mean / standard deviation

Question 6

Discrete vs continuous random variable

Answer 6

Discrete you can count whereas continuous almost always occur when we are measuring

Question 7

Mean of a random variable / expected variable

Answer 7

Mu sub x, described where the probability distribution of X is centered. The mean is an average of all possible values of X, but not all outcomes need to be equally likely.

Question 8

Mean of discrete random variable

Answer 8

Mean equals the sum of the x’s times their probabilities

0(.3)+1(.4)+2(.2)+3(.1) = mean

Question 9

Standard deviation of a random variable

Answer 9

Standard deviation of a random variable, denoted sigma sub x, describes the variability in a probability distribution. When sigma is small, then the observed values of X will be closer to the mean value (little variability) and when sigma is large, there will be more variability (spread) in the observed X values.
Use sigma, not s, because we are not calculating standard deviation of a sample

Question 10

Formula for standard deviation of a secrete random variable

Answer 10

SQRT (sum of (x-mean)^2 (P(X))) = SQRT ((X1-mean)^2 (P(X1)) + (x2-mean)^2 (P(X2)) + …

Question 11

Finding values within one or two standard deviations of mean

Answer 11

Be sure to appropriately “round”
Say interval is (9.03,16.67)–> don’t include 9 or 17 because not in interval
So it is (10,16)

Question 12

Mean of a linear function

Answer 12

Mu sub y = a + b (mu x) where

Note that y = a + bx

Question 13

Standard deviation of linear function

Answer 13

Standard deviation sub y = abs (b) * sigma sub x

Where b is the slope of the linear function

Question 14

Mean and variance of a combination of random variables

Answer 14

If adding them, then you also add the means
If subtracting them, then you also subtract the means

Regardless of adding or subtracting, to get standard deviation –>
SQRT (sigma 1 ^ 2 + sigma 2 ^ 2)
Note: sigma formula only works when X and y are independent. The correlation between two indepdent random variables is zero.

Question 15

How to find mean and standard deviation of a discrete random variable on calculator

Answer 15

Enter values of X in L1 and their probabilities in L2

Then stat, Calc, 1 var stats, then L1 and frequency as L2

Question 16

Binomial requirements

Answer 16

1. There is a fixed number of observations, called trials (n=# of trials)
2. There are only 2 outcomes for each trial, success (S) or failure (F)
3. Outcomes of different trials are independent. That is, knowing the result of one observation tells you nothing about the other observations.
4. The probability of success, p, is the same for each observation.

Question 17

Binomial random variable

Answer 17

Binomial random variable, X, is defined as X = the number of successes observed when the experiment is performed with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n

Question 18

How to write binomial prob distribution

Answer 18

X –> B (n,p)

Question 19

Formula for binomial where n trials, X is a success, and p(x) is probability of success, k is the number of successes

Answer 19

(N choose k) (P(X))^k * (1-P(X))^n-k

Question 20

Binomcdf

Answer 20

This shows for X values less than or equal to
At most K
So say I do binomcdf(10,.2,8) then that’s the probability of X<= 8
For x>=K, then you do 1- binomcdf (n,p,k-1)
For example, at least 7 is binomcdf (10,.2,6)

Question 21

Binomial mean

Answer 21

Mu = n*p

Question 22

Binomial standard deviation

Answer 22

SQRT (n*p(1-p))

Question 23

Normal binomial approximation

Answer 23

If X–> B(n,p) and np >= 10 AND n(1-p)>= 10, then you can use a normal approximation with binomial mean np and standard deviation SQRT (np(1-p))
Then, you convert the value you want to a z score
Then do Normcdf for the values that you want.

Question 24

invNorm

Answer 24

Finds the area to the left of the random variable x

InvNorm (area, mean, sigma)

Question 25

Geometric distribution

Answer 25

Focused on how long it will take to achieve one success

Question 26

Requirements for geometric

Answer 26

1. Trials are performed until a success is achieved
2. There are only 2 outcomes for each trial, success or failure
3. Outcomes of different trials are independent. That is, knowing the result of one observation tells you nothing about the other observations.
4. The probability of success, p, is the same for each observation

Question 27

Geometric random variable

Answer 27

X, defined as, X = number of trials until the first success is observed (including success trial)
X–> G(p)

Question 28

Geometric formula

Answer 28

If X has a geometric distribution with probability p of success and (1-p) of failure on each observation, the possible values of X are 1,2,3,… If n is any one of these values, the probability that the first success occurs on the nth trial is
P(X=n) = (1-p)^n-1 * p, where n is the trial when the first success occurs.
Always skewed right because first trial is the probability and then multiply by decimals

Question 29

Geometcdf

Answer 29

Geometcdf (p,n) shows the probability X<=n

For X>= n, do 1 - geometcdf (p, n-1)

Question 30

Mean of a geometric distribution

Answer 30

Mu = 1/p

Question 31

Standard deviation of geometric random variable

Answer 31

Sigma = SQRT ((1-p)/p^2))

Question 32

Geometric other formula for taking more than a certain number of trials until success:

Answer 32

P(X>n) = (1-p)^n