MATH 1680 - Section 6.1 - Discrete Random Variables

What are the two requirements for a discrete probability​ distribution? Choose the correct answer below. Select all that apply. A. 0≤P(x)≤1 B. 0<P(x)<1 C. ∑P(x)=1 D. ∑P(x)=0

A and C

Which of the following interpretations of the mean is​ correct? A. The observed value of an experiment will be less than the mean of the random variable in most experiments. B. As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable. C. As the number of experiments​ decreases, the mean of the observations will approach the mean of the random variable. D. The observed value of an experiment will be equal to the mean of the random variable in most experiments.

As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable.

Which of the following interpretations of the mean is​ correct? A. If many individuals aged 15 year or older were​ surveyed, the sample mean number of marriages should be close to the mean of the random variable. B. If many randomly selected individuals 40 to 49 years of age were​ surveyed, the sample mean number of marriages should be close to the mean of the random variable. C. If many randomly selected individuals aged 15 years or older were​ surveyed, the observed number of marriages will be less than the mean number of marriages for most individuals. D. If many randomly selected individuals aged 15 years or older were​ surveyed, the observed number of marriages will be equal to the mean number of marriages for most individuals. E. If any number of individuals aged 15 year or older were​ surveyed, the sample mean number of marriages should be close to the mean of the random variable.

If many individuals aged 15 year or older were​ surveyed, the sample mean number of marriages should be close to the mean of the random variable.

random variable

a numerical measure of the outcome to a probability experiment; denoted as X

Suppose we ask a basketball player to shoot three free throws. Let the random variable X represent the number of shots made; so x = 0, 1, 2, or 3. Table 1 shows a probability distribution for the random variable X. Table 1 x P(x) 0 0.01 1 0.10 2 0.38 3 0.51 A) What does the notation P x( ) represent? B) Explain what P( ) 3 0.51 = represents.

a) the probability of the random variable x b) the probability that the basketball player will make three free throws is .51

Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of light bulbs that burn out in the next week in a room with 18 bulbs. ​(b) The time it takes for a light bulb to burn out.

a) the random variable is discrete. 0,1,2,3,4,...,18 b) the random varaible is continious. the possible values are t>0

How to solve: Determine the required value of the missing probability to make the distribution a discrete probability distribution. x P(x) 3 0.18 4 ? 5 0.30 6 0.27

because of the probablity distriubtion rules, the sum of the probabilites must equal one so add all the known probabilities and subtract from one to determine the probability of the random discrete value of four

discrete or continuous random variable: the speed of the next car that is passes a state trooper

continuous

discrete or continuous random variable: the number of As earned in a section of statsitcs with ten studenteds enrolled

discrete

discrete or continuous random variable: the number of cars travel through mcdonald's drive-through in the next hour

discrete

Types of Random Variables

discrete random variable and continuous random variable

for the littery problems, how do you find the expected cash price?

find the mean of the table

for the lottery problems, how do you find the expected profit for one ticket?

find the mean then subract how much you spent on the ticket

Discrete random variables

has finite or countable number of values. in a number line, the values can be plotted with space between each point

A continuous random variable has _______ _______values.

infinitely many

continous Random Variables

inifite amount of values. in a number line, it can be plotted on a line in a uninturrupted fashion

variance of the discrete random variable σ^2x

is the value under the square root in the computaition of the standard deviation

Probability distribution

it gives the possible valeus of a random varaible, X, and its corresponding probabilities -can be denoted through a table, graph, or formula

How to Solve: Determine whether the distribution is a discrete probability distribution.

look ath the table, make sure the probabilites are no smaller than zero nor larger than one and the sum of them are one

9) As the number of repetitions of the experiments increases, what happens to the difference between the mean outcome and the mean of the probability distribution?

the difference of the mean outcome and the mean of the probability distriubtion will get closer to zero as n trials increases

E(X)

the expected value, it's interpretation is the same as the interpretation of the mean of a discrete random variable so if a problem asks for the expected value, compute the mean of the random variable

In the graph of a discrete probability distribution, what do the horizontal axis and the vertical axis represent?

the horizontal axis represents the value of the discrete random variable and vertical axis is teh correspoindg probablity of the discrete random variable

8) As the number of repetitions of the experiments increases, what does the mean value of the n trials approach?

the mean value of n trials will approach the mean of the distribution of the random variable, x

P

the possible value of the random varaible

Rules for a discrete probability distribution

the sum of the probabilities equal one and each probability is between 0 and 1 0 ≤ P(X) ≤ 1, ΣP(X) = 1

how to solve: ​(b) To the nearest​ million, how much should the grand prize be so that you can expect a​ profit?

to find the answer you have to find the grand prize that will yiled the expected profit of zero, then round up to the nearest million so to find the grand prize with zero profit use the expected profit formula: which is the mean minus the cost of the ticket and set it equal to zero bc your profit needs to be zero. then insert the formula of the mean which is the sum of all the multiplication product of x times the p(x). your formula is Σ [x*p(x)]-1=0 the X's will be the cash prize and the p(x) is its probability. you want to find the grand prize so instead of putting in a number for the grandprize, put in x bc ur solving for that. then run up to the nearest million

How to solve: A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one-year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability that the female will survive the year is 0.99791. Compute the expected value of this policy to the insurance company.

to solve this theres two x values. the first one is the cost of the one year insurance (eg $530) and the second x value is the first one minus the toatal price of the insurance (eg $250,000). the two x values are 530 and -249,470. the p(x) of the first one is .99791 and the p(x) of the second value is .00209. to find the expected value of the insurance company, compute the mean by going to the costum calculator on stat crunch. the answer would be $7.50. the interpretation: on average, for each person ensurance the insurance company expects to make $7.50

6) When graphing a discrete probability distribution, how do we emphasize that the data is discrete?

we draw the graph of a _______ __________ _______ using vertical lines above each value of the random variable to a height that is the probability of the random variable

formula for the mean of a discrete random variable

Σ [ xi * P(xi) ] multiply each X by its probability and add all of them together

formula for computing the standard deviation of a discrete random variable

√Σ [ (x-μx)^2 * P(x) ] add products of multimplying (the value of the random variable - the mean of the random variable)^2 by the proababilty of the random variable then find the square root of the sum