Category A
This takes a fixed set of possible numerical values that describe the outcomes of some chance process.
Discrete Random Variable
Gives possible values and their probabilities for a random variable
probability distribution
This is also know as an expected value of a discrete random variable and is found by multiplying each possible value by it's probability and then adding up all the products.
Mean of a discrete random variable
This is the average of the squared deviation of the values of a random variable from it's mean, and like the mean this value is weighted by the probabilty that the value will occur.
Variance of a discrete random variable.
this takes all values in an interal of numbers
This takes all values in an interval of numbers of that describe the outcomes of some chance process.
continuous random variable
This causes the values of center and location to change by a factor of b, and causes the measures of spread to be changed by a factor of |b|, but does not change the shape of a distribution.
Effect of a random variable being multiplied or divided by a constant.
This causes the values of center and location to change by shifting in the direction and amount of a, but does not change the measures of spread to be or the shape of a distribution.
Effect on a random variable of adding/subtracting a constant.
This happens to mean if any two random variables are added together
The mean of each variable is added together to creat the transformed mean
This happens to the variance when any two independent random variables are added together
the variance of each variable is added together to create the transformed variance
The sum or difference of independent Normal random variables follows this type of distribution.
Normal distribution
These conditions must be met to confirm a binomial setting exists.
Binary - outcomes of each trial can be success or failure.
Independent - each trial is independent.
Numbers - there is a set number of trials.
Success - on each trial probabilty of success is the same.
This is the count of successe in a binomial setting.
Binomial Random Variable
The calculation used to find the probability of a specific value (k) out of (n) trials for a binomial distribution.
P(X=k)=((n),(k))p^k(1-p)^(n-k)
This is how the mean of a binomial random variable is calculated.
mu_{x}=np
This is how the standard deviation of a binomail random variable is caclculated.
sigma_{x}=sqrt(np(1-p)
When this condition is met, you can use a binomial distribution model to model the count of successes in a SRS of size (n).
The population is at least 10 times the sample size.
n<=1/10N
These conditions must be met to confirm a geometric setting exists.
Binary - outcomes of each trial can be success or failure.
Independent - each trial is independent.
Trials - the number of trials is counted until the first success occurs.
Success - on each trial probabilty of success is the same.
This is how a geometric probability is calculated.
P(Y=k)=(1-p)^(k-1)p
If Y is a geometric random variable with probability of success p on each trial, then its mean is found by doing this. (The expected number of trials required to get the first success.)
1/p
This is the primary difference in distiguishing between a binomial and a geometric setting.
Binomial is has a set number of trials and counts the number of successes where Geometric counts the number of trials until the first success.
A marketing survey compiled data on the number of cars in households. If X=the number of cars in a randomly selected household, and we omit the rare cases of more than 5 cars, then X has the following probability distribution:
x 0 1 2 3 4 5
P(x) 0.24 0.37 0.20 0.11 0.05 0.03
The probability that a radomly chosen household has at least two cars.
Add all probabilities for each value of x that considered a "success"
0.20+0.11+0.05+0.03 = 0.39
A marketing survey compiled data on the number of cars in households. If X=the number of cars in a randomly selected household, and we omit the rare cases of more than 5 cars, then X has the following probability distribution:
x 0 1 2 3 4 5
P(x) 0.24 0.37 0.20 0.11 0.05 0.03
The expected value of the number of cars in a randomly selected household is:
Find the weighted average. Take the value times the probability that value will occur and add up all of the products.
0(0.24)+1(0.37)+2(0.20)+3(0.11)+4(0.05)+5(0.03)=1.45
A dealer in Las Vegas selects 10 cards from a standard deck of 52 cards. Let Y be the number of diamonds in the 10 cards selected. List the type of distribution, the number of trials and probability of this scenario along with any presumptions.
Presuming the cards are replaced and shuffled between each trial, this would be a binomial distribution (set number of independent trials, each with the same probability of success and number of successes is counted) The sample size is n=10, and the probability is p=0.25
In the town of lakevill, the nuber of cell phones in a household is a random variable W with the following probability distribution.
Value (W) 0 1 2 3 4 5
Probability (P) 0.1 0.1 0.25 0.3 0.2 0.05
The standard deviation of the number of cell phones in a randomly selected house:
To calculate the standard deviation you need to first calculate the mean by multiplying each value of W by it's probability, then add up results for each. Once you have the mean, you calculate the variance. Take each value of W, deduct the mean, and square the result, finally sum each result. Now to convert the variance to the standard deviation, you squareroot the variance.
Mean:
(0)(.1)+.....+(5)(.05)=2.55
Variance:
(0-2.55)^2+....(5-2.55)^2=1.7475
Standard Deviation:
sqrt(1.7475)=1.32193
A random variable Y has the following probability distribution:
Value (Y) -1 0 1 2
P(Y) 4C 2C 0.07 0.03
The value of the constant C is:
The total probability must add up to 1 (or 100%).
4C+2C+0.07+0.03=1.00
6C+.1=1.00
C=.15
It is known that about 90% of the widgets made by Buckley Industries meet specifications. Every hour a sample of 18 widgets is selected at random for testing and the number of widgets that meet specifications is recorded. What is the approximate mean and standard deviation of the number of widgets meeting specifciations.
The mean of a random binomial variable is calculated by multiplying the sample size to the probability of "success" and the standard deviation is calculated by multiplying the number of successes by the probability of success and then 1 minus the probability of success.
Mean:
μ x = (18)(.9)=16.2
Standard Deviation:
sqrt(18(.9)(1-.9))=1.273
The variance of the sum of two random variables X Y is:
When you add two random variables, the variance of the transformation is also added.
sigma_{X}+sigma_{Y}
A raffle sells tickets for $10 and offers a prize of $500, $1000, or $2000. Let C be a random variable that represents the prize in the raffle drawing. The probability distribution of C is given below:
C $0 $500 $1000 $2000
P(C) 0.60 0.05 0.13 0.11
The expected profit when playing the raffle is:
Expected value is the same thing as the mean, so you calculate the expected profit by multiplying the value of C times the probability of C, and summing up the products.
(0)(.6)+(500)(.05)+(1000)(.13)+(2000)(.11)=595
Let the random variable X represent the amount of money Carl makes tutoring statistics students in the sumer. Assume that X is Normal with mean $240 and standard deviation $60. The probability is approximately 0.6 that, in a randomly selected summer, Carl will make less than about:
To calculate the value of X that has a 60% probability, you can use Table A to find the corresponding z-score and then use the mean and standard deviation to find the X value. You can also use the inverse norm function on the TI-84.
2nd -> distr ->invNorm(.6,240,60,LEFT) Paste, ENTER = 255.20
Describe which of the following is a geometric random variable and why:
a. The number of phone calls received in a 1-hour period.
b. The number of times I have to roll a 6-sied die to get two fives.
c. The number of digits I will read beginning at a randomly selected starting point in a table of random digits until I find a 7.
d. The number of sevens in a row of 40 random digits.
E. All four of the above are geometric random variables.
Option C, "the number of digits I will read beginning at a randomly selected starting point in a table of random digits until I find a 7".
A geometric random variable scenario must meet the following criteria:
Binary - to be considered a "success" the digit found would be 7, anything else would be "failure".
Independent - each digit chosen is independent of the next.
Trials - the number of trials is counted until the first 7 is read.
Success - on each trial, the probabilty of success is the same.