Key

Concept

  1. Distributions of Random Variables - A rule that assigns a number to each outcome of an experiment.
  2. Identify the type of Random variable
  3. Finite Discrete Random variable
  4. Infinitely Discrete Random variable
  5. Continuous Random Variable
  6. Identify the X
  7. Find the number of outcomes and find the values assigned to each outcome of the experiments by a random variable of X
  8. Find the outcomes of the events
  9. find the probabilities of each distributions

Example:

-A Coin tossed 4 times and the # of heads that occur in the 4 tosses is recorded.

-X= # of heads

a.) list the outcomes of the experiment and find the value assigned to each outcome of the experiment by the random variable x.

Outcome / HHHH / HHHT / HHTT / HTHT / HTTH / THHT / HTTT / TTHT / TTTH / HHTH
x / 4 / 3 / 2 / 2 / 2 / 2 / 1 / 1 / 1 / 3
outcome / HTHH / THHH / THTT / THTH / TTHH / TTTT
x / 3 / 3 / 1 / 2 / 2 / 0

b.) Find outcomes X=3

HHHT, HHTH, HTHH, THHH

c.) Find the probability distribution for the data

x / 0 / 1 / 2 / 3 / 4
P(X=x) / 1/16 / 4/16 / 6/16 / 4/16 / 1/16

2. Histograms - A bar graph with no space, each bar has a width of 1 area, and each

  1. Find the x and y
  2. x is the value of the random variables
  3. y is the probability of an outcome
  4. graph it

Example: Draw a histogram based on the data from example 1

1/16 4/16 6/16 4/16 1/1

3. Expected value- values of X associated with the probabilities (P)

to find the expected profit:

1)Expected profit= amount of payment( probability) +

Total amount - payment ( odds of probability )

Example: Students from a small college were asked how many cards they carry. X is the random variable representing the number of cards and the results are below. Find the average number of cards carried by college students at the college.

X / 0 / 1 / 2 / 3 / 4 / 5 / 6
# people / 13 / 43 / 58 / 25 / 10 / 5 / 3

Total of Cards/ # of people =

0(13)+1(43)+2(58)+3(25)+4(10)+5(5)+6(3)/157= 2.02

4. Standard deviation-a measure that is used to quantify the amount of variation or dispersion of a set of data values.

You and your friends have just measured the heights of your dogs (in millimeters):The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Your first step is to find the Mean:

Answer:

Mean = / 600 + 470 + 170 + 430 + 300 / = / 1970 / = 394
5 / 5

so the mean (average) height is 394 mm. Let's plot this on the chart:

5. Variance- a measure of the spread of the data

(-)+...... +(- )

Example: Find the Variance of the weights of brand X and brand Y Doritos. (from problem 4)

To calculate the Variance, take each difference, square it, and then average the result:

So, the Variance is 21,704. And the Standard Deviation is just the square root of Variance, so: Standard Deviation: σ = √21,704 = 147.32... = 147 (to the nearest mm) And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation (147mm) of the Mean:

So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small. Rottweilers are tall dogs. And Dachshunds are a bit short ... but don't tell them!

6. Binomial trials- probability of success

C(n,x)

n= # of trials

x= # of successes

P= Probability of success

Q= Probability of failure

Standard deviation of random variable X

Binomialpdf(n,p,x)

Example: From experience, the manager at a bookstore knows that 40% of the people are browsing the store will make a purchase. What is the probability that among the 10 people, at least 3 people will make a purchase?

n=10

x=0,1,2

p=.4

B0+B1+B2= .1678897536

1-.1678897536= .833

7. Z- score- number of standard deviations away from the mean on the normal distribution

Example:

a.) What is the percent that falls within the standard deviation? =68%

b.) What is the percent that falls within the standard deviation? =95%

c.)What is the percent that falls within the standard deviation? =99.7%

8. Normal distributions- probabilities that aren't in a normal distribution

Normalcdf(lower bound, upper bound, , )

Calculator:2nd, vars, 2

InvNorm( percentage of area to the left, mean, standard deviation)

Calculator: 2nd, vars, 3

Example: 1000 Freshman at State University took a Biology est. The scores were distributed normally with a mean of 60 and a standard deviation of 6. Label the mean and three standard deviations from the mean.

42 48 54 60 64 68 72

a.) What percentage of scores are between scores 60 and 72?

50% =(.34)(.135)(.0235)(.0015)

b.) What percentage of scores are between scores 54 and 60?

34% =(.34)

c.) What percentage of scores are between 42 and 64?

84% =(.0015)(.0235)(.135)(.34)(.34)

d.) Approximately how many biology students scored between 42 and 64?

840 students =(.84)1000

Example: 500 Juniors at Central high took the ACT last year. The scores were distributed normally with a mean of 25 and a standard deviation of 5. Label the mean and three standard deviations from the mean.

10 15 20 25 30 35 40

a.) What percentage of scores are between scores 10 and 25?

50% =(.0015)(.0235)(.135)(.34)

b.) What percentage of scores are between scores 35 and 40?

.15% =(.0015)(100)

c.) What percentage of scores are between scores 30 and 40?

.16% =(.135)(.0235)(.0015)

d.) What percentage of scores are between scores 25 and 35?

47.5% =(.34)(.135)

9. Simulations- A way to model random events to closely match real world outcomes (This saves time, effort, or money)

RandInt (min. Value, max Value,# of data in set)

Calculator: to find randInt

1)math

2)PRB

3)1

Example: Can Molly roll one of each number in a throw of six?

-rantlnt (1,6,6)

1 / x / 6 / x / 11 / x / 16 / x
2 / ✔ / 7 / x / 12 / x / 17 / x
3 / x / 8 / x / 13 / x / 18 / x
4 / x / 9 / x / 14 / x / 19 / x
5 / x / 10 / x / 15 / x / 20 / x

Results = 1/20 5% chance that Molly will succeed

Vocabulary[1]

Finite Discrete- random variable has a finite number of values. Ex: # of heads in four tosses.

Infinitely Discrete- random variable has infinitely many values. Ex: # of tosses before heads appears

Continuous- random variable has values which are in an interval of real numbers. Ex: amount of time (in minutes) it takes a preschooler to tie his shoe.

Histogram- graphical representation of the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable

Ex:

Mean- (average) the sum of a collection of numbers divided by the number of numbers in the collection

Odds- a numerical expression, usually expressed as a pair of numbers, used in both gambling and statistics Ex: if P(E) is the probability of an event E occurring, then: odd in favor of E = P(E)/P(EC), and odds against E = P(EC)/P(E)

Variance- a measure of the spread of the data (the larger the variance, the larger the spread.) Ex: Var(X) = p1(x1)2 + p2(x2)2 + … + pn (xn)2

Standard Deviation- a measure of the spread of the data using the same units as the data. Ex: or

Parameter- an entire population Ex:

Statistic- a sample of the population

Z-score- the number of standard deviations () a value is from the mean

(

Empirical Rule- nearly all values lie with in three standard deviations of the mean in a normal distribution

[1]if you want to move or change around the vocab; go for, just let me know what else you guys want me to do