Lecture 4 Flashcards
What is a random variable?
Numerical value associated with each outcome represented by x
Discrete Random Variable
Has a finite (countable) # of options
ex. number of shirts you sell
Continuous random variable is….
an uncountable number of possible outcomes,
represented by an interval on a number line (e.g., time spent on the phone calls)
Includes decimals
The number (x) of Fortune 500 companies that
lost money in the previous year
discrete
Geometric distribution
A discrete probability distribution
(slightly different than binomial distribution). Must satisfies the
following conditions:
1. A trial is repeated until a success occurs (e.g., flip a coin until you get a heads).
2. The repeated trials are independent of each other.
3. The probability of success p is constant for each trial.
4. The random variable x represents the number of the trial in
which the first success occurs.
❑ The probability that the first success will occur on trial x is:
P(x) = p1(q)x – 1, where q = 1 – p.
NOTE: Zero is not a possible outcome
here. Dependent on at least one success.
Which of the following scenarios explains a geometric distribution?
I. Seed Depot advertises that 85% of flower seeds will germinate (grow). Suppose that the company’s claim is true. Judy buys a packet with 20 flower seeds from Seed Depot and plants them in her garden. Let x = the number of seeds that germinate.
II. Put the names of all the students in the class in a hat. Mix them up, and draw four names without looking. Let y = the number whose last names have more than 6 letters.
III. Exactly 10% of the students in this school are left-handed. Select students at random from the school, one at a time, until you find one who is left handed. Let v = the number of students chosen.
IV. Exactly 10% of the students in this school are left-handed. Select 15 students at random from the school and define w = the number of students who are left handed.
lll.
A worn out bottling machine does not properly apply caps to 5% of the bottles it fills. In a production run of 800 bottles, what is the standard deviation for the number of bottles with improperly applied caps?
This is a binomial distribution since the machine either properly caps the bottles or does not cap them properly. The formula for the standard deviation of a binomial distribution is
σ=√n⋅p⋅(1−p)
Our “n” in this problem is 800 bottles, and the “p” in this problem is 0.05. So, we will multiply 800 × 0.05 × (1-0.05) to get 38. Then, we take the square root of 38 to get the answer of 6.16.
What is the formula for sd of a binomial distribution
What’s a binomial distribution?
σ=√n⋅p⋅(1−p)
Binomial is: probability of exactly successes on repeated trials in an experiment which has two possible outcomes
A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000 profit. If it is very cold that day, she expects to suffer a $12,000 loss. Based upon historical records, the weather office has estimated chances of a warm day to be 0.60; the chances of a cool day to be 0.25. What is the producer’s expected profit?
11450
In a certain large population, 40% of the households have a total annual income of over $70,000. A simple random sample of 4 of these households is selected. Let the number of households in the sample with an annual income of over $70,000 be our random variable and assume that the binomial assumptions are reasonable. What is the mean?
The mean of a binomial distribution is the total number of trials multiplied by the probability of the certain outcome. In this case, there are a total of 4 households surveyed, and the probability of one of them making over $70,000 is 0.40. So, we multiply 4 by 0.40 to get 1.6.
Which of the following is a true statement?
I. The binomial setting requires that there are only two possible outcomes for each trial, while the geometric setting permits more than two outcomes.
II. A geometric random variable takes on integer values from 0 to n.
III. If X is a geometric random variable and the probability of success is 0.85, then the probability distribution of X will be skewed left, since 0.85 is closer to 1 than to 0.
IV. An important difference between binomial and geometric random variables is that there is a fixed number of trials in a binomial setting, and the number of trials varies in a geometric setting.
IV
For statement I, both binomial and geometric distributions require only two possible outcomes for each event. For statement II, a geometric distribution cannot take on a value of 0. There has to be at least one event in a geometric distrubution. The distribution in part III would be skewed right. Statement IV is correct since a binomial distribution has a fixed number of trials while the geometric distribution does not have the fixed number. That is the main difference between the two distributions. Geometric distributions look for the trial when the first success happens.
Which of the following are binomial distributions?
I. An inspection procedure at an automobile manufacturing plant involves selecting a sample of cars from the assembly line and noting for each car whether there are no defects, at least one major defect, or only minor defects.
II. As students study more and more during their Statistic class, their chances of getting an A on any given test continue to improve. The teacher is interested in the probability of any given student receiving various numbers of A’s on the class exams.
III. A committee of two is to be selected from among the five teachers and ten students attending a meeting. A researcher is interested in finding the probability that the committee will consist of exactly one teacher and one student.
NONE
binomial experiements:
The experiment is repeated for a fixed number of trials,
where each trial is independent of other trials.
2. There are only two possible outcomes of interest for each
trial. The outcomes can be classified as a success (S) or as a
failure (F).
3. The probability of a success, P(S), is the same for each trial.
4. The random variable x counts the number of successful
trials.
.
Example: Draw a card from standard deck, note
whether it is a club or not, and replace the card.
Do this 5 times.
Rules of a chart showing prob dis
For a chart to show a valid probability distribution, a few requirements need to be met. First, each individual probability value must be a number between 0 and 1 inclusive. Next, each outcome must be a numeric random variable. Since all of the X values are numbers, this requirement is also met. The last requirement is that all of the probabilities must add up to exactly 1.0. When you add up 0.09 + 0.36 + 0.49 + 0.10, you get 1.04. Since this answer is over 1.0, it is not a valid probability distribution. There must have been some error in collecting the data that made the results come out to over 100%.
Twenty percent of all trucks undergoing a certain inspection will fail the inspection. Assume that trucks are independently undergoing this inspection, one at a time. What is expected number of trucks inspected before a truck fails inspection?
This is an example of a geometric distribution since we are looking for the number of trials before we get the first truck to fail inspection. The mean of a geometric distribution is one divided by the probability of event occurring. Since 20% of all trucks fail inspection, we will take one divided by 0.20 which is 5. So, we can expect to have to inspect 5 trucks before we get one that fails the inspection.
A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52 cards. Let Y be the number of red cards (hearts or diamonds) in the 40 cards selected. Which of the following best describes this setting:
I. Y has a binomial distribution with n = 40 observations and probability of success p = 0.5.
II. Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided the deck is shuffled well.
III. Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided after selecting a card it is replaced in the deck and the deck is shuffled well before the next card is selected.
IV. Y has a geometric distribution with n = 40 and p = 0.5.
lll
Probability distribution
What are all the possible outcomes?
To calculate discrete probability distribution what do we do
Take number /tot
To calculate mean of discrete probability distribution you must
Multiply outcomes with x
Variance of discrete probability distribution
σ² = ∑ (x - μ)² * P(x)
SD of discrete probability distribution
σ = √(∑ (x - μ)² * P(x))
Find the mean first: Before calculating the standard deviation, you need to calculate the mean (expected value) of the discrete distribution using the formula: μ = ∑ (x * P(x)).
Calculate deviations from the mean: For each possible value (x), subtract the mean (μ) to get the deviation (x - μ).
Square the deviations: Square each deviation (x - μ)².
Multiply by probabilities: Multiply each squared deviation by its corresponding probability P(x).
Sum the products: Add up all the products from the previous step.
Take the square root: Finally, take the square root of the sum to get the standard deviation (σ).
Expected value of a discrete random variable
= to the mean of the random x variable
Its the value you get when you do a tone of trails
If a game is fair expected value is 0
If not, it might be negative
Binomial experiments
- The experiment is repeated for a fixed number of trials,
where each trial is independent of other trials. - There are only two possible outcomes of interest for each
trial. The outcomes can be classified as a success (S) or as a failure (F). - The probability of a success, P(S), is the same for each trial.
- The random variable x counts the number of successful
trials.
Example: Draw a card from standard deck, note
whether it is a club or not, and replace the card.
Do this 5 times.
Is this a binomial experiment?If it is, specify the values of
n (trails), p(success), and q (prob of fail), and list the possible values of the random variable x. If not why?
Draw a card from standard deck, note
whether it is a club or not, and replace the card.
Do this 5 times.
yes
Binomial table use it when….
to determine the probability of a specific event happening
Determine whether the statement is true or false. If it is false, rewrite it as a true statement.
In most applications, continuous random variables represent counted data, while discrete random variables represent measured data.
false
What is a discrete probability distribution? What are the two conditions that determine a probability distribution?
Question content area bottom
Part 1
What is a discrete probability distribution? Choose the correct answer below.
A.
A discrete probability distribution lists each possible value a random variable can assume, together with its probability.
B.
A discrete probability distribution lists each possible value a random variable can assume.
C.
A discrete probability distribution exclusively lists probabilities.
D.
None of the above
a
What are the two conditions that determine a probability distribution? Choose the correct answer below.
A.
The probability of each value of the discrete random variable is between 0 and 1, inclusive, and the sum of all the probabilities can be any amount.
B.
The probability of each value of the discrete random variable is greater than 0 and less than 1, and the sum of all the probabilities is 1.
C.
The probability of each value of the discrete random variable is between 0 and 1, inclusive, and the sum of all the probabilities is 1.
D.
The probability of each value of the discrete random variable is greater than 0 and less than 1, and the sum of all the probabilities can be any amount.
c
Determine whether the distribution is a discrete probability distribution.
x P(x)
0 0.25
1 0.30
2 -0.10
3 0.30
4 0.25
Is the distribution a discrete probability distribution? Why? Choose the correct answer below.
A.Yes comma because the distribution is symmetric.
Yes comma because the distribution is symmetric.
B.Yes comma because the probabilities sum to 1 and are all between 0 and 1 comma inclusive.
Yes comma because the probabilities sum to 1 and are all between 0 and 1 comma inclusive.
C.No comma because some of the probabilities have values greater than 1 or less than 0.
No comma because some of the probabilities have values greater than 1 or less than 0.
D.No comma because the total probability is not equal to 1.
No comma because the total probability is not equal to 1.
c can’t have - probability
The expected value of an accountant’s profit and loss analysis is 0. Explain what this means.
Question content area bottom
Part 1
Choose the correct answer below.
A.
Since the expected value cannot be less than 0, an expected value of 0 means that the average money gained is equal to or less than the average money spent.
B.
An expected value cannot be equal to 0.
C.
An expected value of 0 means that there was not any money gained or spent.
D.
An expected value of 0 means that the average money gained is equal to the average money spent, representing the break-even point.
Less than …. is unusual
0.05
N=
P=
Q=
Sucess=
X=
N= # trials
P= %
Q= # fails
Success=
X= possible values of random x
Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.
n equals 90, p equals 0.4
The variance of a binomial distribution is given by
sigma squared equals npq
where n is the number of times a trial is repeated, p is the probability of success in a single trial, and q is the probability of failure in a single trial (q =1 - p).
The mean=36
The variance=21.6
The standard deviation=4.6
The variance of a binomial distribution is given by
sigma squared equals …..
npq
In a geometric distribution, the probability that the first success will occur on trial number x is given by the following equation, where p is the probability of a success and q equals 1 minus p.
P(x) = p(q)^x – 1
