AP Stat Ch 6 Flashcards
Discrete random variable
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.
Random variable
A numerical variable whose value depends on the outcome of a chance experience.
Properties of a discrete random variable, X
0<=P(X)<=1
Sum of P(X)=1
Continuous random variable
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.
Properties of density functions
- f(x)>= 0
- Total area under curve = 1
- Probability that x falls in any particular interval is e area under curve in that interval
How to calculate z score
Z = X- mean / standard deviation
Discrete vs continuous random variable
Discrete you can count whereas continuous almost always occur when we are measuring
Mean of a random variable / expected variable
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.
Mean of discrete random variable
Mean equals the sum of the x’s times their probabilities
0(.3)+1(.4)+2(.2)+3(.1) = mean
Standard deviation of a random variable
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
Formula for standard deviation of a secrete random variable
SQRT (sum of (x-mean)^2 (P(X))) = SQRT ((X1-mean)^2 (P(X1)) + (x2-mean)^2 (P(X2)) + …
Finding values within one or two standard deviations of mean
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)
Mean of a linear function
Mu sub y = a + b (mu x) where
Note that y = a + bx
Standard deviation of linear function
Standard deviation sub y = abs (b) * sigma sub x
Where b is the slope of the linear function
Mean and variance of a combination of random variables
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.
How to find mean and standard deviation of a discrete random variable on calculator
Enter values of X in L1 and their probabilities in L2
Then stat, Calc, 1 var stats, then L1 and frequency as L2
Binomial requirements
- There is a fixed number of observations, called trials (n=# of trials)
- There are only 2 outcomes for each trial, success (S) or failure (F)
- Outcomes of different trials are independent. That is, knowing the result of one observation tells you nothing about the other observations.
- The probability of success, p, is the same for each observation.
Binomial random variable
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
How to write binomial prob distribution
X –> B (n,p)
Formula for binomial where n trials, X is a success, and p(x) is probability of success, k is the number of successes
(N choose k) (P(X))^k * (1-P(X))^n-k
Binomcdf
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)
Binomial mean
Mu = n*p
Binomial standard deviation
SQRT (n*p(1-p))
Normal binomial approximation
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.
invNorm
Finds the area to the left of the random variable x
InvNorm (area, mean, sigma)
Geometric distribution
Focused on how long it will take to achieve one success
Requirements for geometric
- Trials are performed until a success is achieved
- There are only 2 outcomes for each trial, success or failure
- Outcomes of different trials are independent. That is, knowing the result of one observation tells you nothing about the other observations.
- The probability of success, p, is the same for each observation
Geometric random variable
X, defined as, X = number of trials until the first success is observed (including success trial)
X–> G(p)
Geometric formula
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
Geometcdf
Geometcdf (p,n) shows the probability X<=n
For X>= n, do 1 - geometcdf (p, n-1)
Mean of a geometric distribution
Mu = 1/p
Standard deviation of geometric random variable
Sigma = SQRT ((1-p)/p^2))
Geometric other formula for taking more than a certain number of trials until success:
P(X>n) = (1-p)^n
