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# Discrete Probability Distribution Questions And Answers Pdf

By  Axel C.

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The idea of a random variable can be confusing.

## 4.2: Probability Distribution Function (PDF) for a Discrete Random Variable

Associated to each possible value x of a discrete random variable X is the probability P x that X will take the value x in one trial of the experiment. The probability distribution A list of each possible value and its probability. The probabilities in the probability distribution of a random variable X must satisfy the following two conditions:.

A fair coin is tossed twice. Let X be the number of heads that are observed. The possible values that X can take are 0, 1, and 2. The probability of each of these events, hence of the corresponding value of X , can be found simply by counting, to give.

This table is the probability distribution of X. A histogram that graphically illustrates the probability distribution is given in Figure 4. Figure 4. A pair of fair dice is rolled. Let X denote the sum of the number of dots on the top faces. The possible values for X are the numbers 2 through Continuing this way we obtain the table.

Before we immediately jump to the conclusion that the probability that X takes an even value must be 0. We compute.

The mean of a random variable may be interpreted as the average of the values assumed by the random variable in repeated trials of the experiment. Find the mean of the discrete random variable X whose probability distribution is. A service organization in a large town organizes a raffle each month. Each has an equal chance of winning. Let X denote the net gain from the purchase of one ticket. Let W denote the event that a ticket is selected to win one of the prizes.

Using the table. The negative value means that one loses money on the average. In particular, if someone were to buy tickets repeatedly, then although he would win now and then, on average he would lose 40 cents per ticket purchased. The concept of expected value is also basic to the insurance industry, as the following simplified example illustrates.

Find the expected value to the company of a single policy if a person in this risk group has a Let X denote the net gain to the company from the sale of one such policy. There are two possibilities: the insured person lives the whole year or the insured person dies before the year is up.

Since the probability in the first case is 0. The variance and standard deviation of a discrete random variable X may be interpreted as measures of the variability of the values assumed by the random variable in repeated trials of the experiment. The units on the standard deviation match those of X. A discrete random variable X has the following probability distribution:. Determine whether or not the table is a valid probability distribution of a discrete random variable.

Explain fully. Borachio works in an automotive tire factory. The number X of sound but blemished tires that he produces on a random day has the probability distribution. In a hamster breeder's experience the number X of live pups in a litter of a female not over twelve months in age who has not borne a litter in the past six weeks has the probability distribution. The number X of days in the summer months that a construction crew cannot work because of the weather has the probability distribution.

Let X denote the number of boys in a randomly selected three-child family. Assuming that boys and girls are equally likely, construct the probability distribution of X. Let X denote the number of times a fair coin lands heads in three tosses. Construct the probability distribution of X. Let X denote the net gain from the purchase of a randomly selected ticket. An insurance company estimates that the probability that an individual in a particular risk group will survive one year is 0.

Let C denote how much the insurance company charges such a person for such a policy. A roulette wheel has 38 slots. Thirty-six slots are numbered from 1 to 36; half of them are red and half are black. The remaining two slots are numbered 0 and 00 and are green. If the ball lands in a red slot, he receives back the dollar he bet plus an additional dollar. If the ball does not land on red he loses his dollar. Let X denote the net gain to the bettor on one play of the game. Thirty-six slots are numbered from 1 to 36; the remaining two slots are numbered 0 and If the ball lands in an even numbered slot, he receives back the dollar he bet plus an additional dollar.

If the ball does not land on an even numbered slot, he loses his dollar. The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. As sometimes happens with probabilities computed as empirical relative frequencies, probabilities in the table add up only to a value other than 1.

Tybalt receives in the mail an offer to enter a national sweepstakes. If it costs Tybalt 44 cents to mail his entry, what is the expected value of the sweepstakes to him? The number X of nails in a randomly selected 1-pound box has the probability distribution shown.

Find the average number of nails per pound. Three fair dice are rolled at once. Let X denote the number of dice that land with the same number of dots on top as at least one other die. The probability distribution for X is.

Two fair dice are rolled at once. Let X denote the difference in the number of dots that appear on the top faces of the two dice. A fair coin is tossed repeatedly until either it lands heads or a total of five tosses have been made, whichever comes first. Let X denote the number of tosses made.

A manufacturer receives a certain component from a supplier in shipments of units. Two units in each shipment are selected at random and tested. If either one of the units is defective the shipment is rejected.

Suppose a shipment has 5 defective units. Shylock enters a local branch bank at p. The number X of customers in the bank who are either at a teller window or are waiting in a single line for the next available teller has the following probability distribution.

The owner of a proposed outdoor theater must decide whether to include a cover that will allow shows to be performed in all weather conditions. Based on projected audience sizes and weather conditions, the probability distribution for the revenue X per night if the cover is not installed is.

The owner will have it built if this cost can be recovered from the increased revenue the cover affords in the first ten night seasons. Previous Section. Table of Contents. Next Section. To learn the concepts of the mean, variance, and standard deviation of a discrete random variable, and how to compute them.

Probability Distributions Associated to each possible value x of a discrete random variable X is the probability P x that X will take the value x in one trial of the experiment. Definition The probability distribution A list of each possible value and its probability.

Example 1 A fair coin is tossed twice. Find the probability that at least one head is observed. Solution: The possible values that X can take are 0, 1, and 2. The probability of each of these events, hence of the corresponding value of X , can be found simply by counting, to give x 0 1 2 P x 0. Example 2 A pair of fair dice is rolled. Find the probability that X takes an even value. Solution: The sample space of equally likely outcomes is 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66 The possible values for X are the numbers 2 through Continuing this way we obtain the table x 2 3 4 5 6 7 8 9 10 11 12 P x 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 This table is the probability distribution of X.

Example 4 A service organization in a large town organizes a raffle each month. Find the probability of winning any money in the purchase of one ticket. Find the expected value of X , and interpret its meaning.

Solution: Let X denote the net gain to the company from the sale of one such policy. Key Takeaways The probability distribution of a discrete random variable X is a listing of each possible value x taken by X along with the probability P x that X takes that value in one trial of the experiment.

Exercises Basic Determine whether or not the table is a valid probability distribution of a discrete random variable. A discrete random variable X has the following probability distribution: x 77 78 79 80 81 P x 0. A discrete random variable X has the following probability distribution: x 13 18 20 24 27 P x 0.

## Discrete probability distributions

There are two types of random variables , discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. These values are obtained by measuring by a thermometer.

## Probability Distributions: Discrete and Continuous

A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a hour shift. For a random sample of 50 patients, the following information was obtained.

In probability theory and statistics , a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. For instance, if X is used to denote the outcome of a coin toss "the experiment" , then the probability distribution of X would take the value 0. Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey , etc.

#### Concept Review

Sign in. Random Variables play a vital role in probability distributions and also serve as the base for Probability distributions. Before we start I would highly recommend you to go through the blog — understanding of random variables for understanding the basics. Today, this blog post will help you to get the basics and need of probability distributions. What is Probability Distribution? Probability Distribution is a statistical function which links or lists all the possible outcomes a random variable can take, in any random process, with its corresponding probability of occurrence. Values o f random variable changes, based on the underlying probability distribution.

We now define the concept of probability distributions for discrete random variables, i. Such random variables generally take a finite set of values heads or tails, people who live in London, scores on an IQ test , but they can also include random variables that take a countable set of values 0, 1, 2, 3, …. A frequency function can be expressed as a table or a bar chart, as described in the following example. Example 1 : Find the distribution function for the frequency function given in columns A and B below. Also show the graph of the frequency and distribution functions. Figure 1 — Table of frequency and distribution functions. Using the approach described in Example 2.

A random variable is a numerical description of the outcome of a statistical experiment. A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete; one that may assume any value in some interval on the real number line is said to be continuous. For instance, a random variable representing the number of automobiles sold at a particular dealership on one day would be discrete, while a random variable representing the weight of a person in kilograms or pounds would be continuous. The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x , the probability distribution is defined by a probability mass function, denoted by f x. This function provides the probability for each value of the random variable. In the development of the probability function for a discrete random variable, two conditions must be satisfied: 1 f x must be nonnegative for each value of the random variable, and 2 the sum of the probabilities for each value of the random variable must equal one.

There are two types of random variables , discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables.

Беккер мрачно оглядел море красно-бело-синих причесок. - Что у них с волосами? - превозмогая боль, спросил он, показывая рукой на остальных пассажиров.  - Они все… - Красно-бело-синие? - подсказал парень. Беккер кивнул, стараясь не смотреть на серебряную дужку в верхней губе парня. - Табу Иуда, - произнес тот как ни в чем не бывало.

Будь он менее самонадеян, он, конечно же, заглянул бы в энциклопедию и обнаружил, что это не что иное, как солевой осадок, оставшийся после высыхания древних морей. Как и все криптографы АНБ, Хейл зарабатывал огромные деньги, однако вовсе не стремился держать этот факт при. Он ездил на белом лотосе с люком на крыше и звуковой системой с мощными динамиками.

Сьюзан, - умоляюще произнес Стратмор, не выпуская ее из рук.  - Я все объясню. Она попыталась высвободиться. Коммандер не отпускал. Она попробовала закричать, но голос ей не повиновался.

Изначальный план состоял в том, чтобы сделать это незаметно и позволить Танкадо продать пароль. Сьюзан должна была признать, что прозвучало это довольно убедительно. У Танкадо не было причин подозревать, что код в Интернете не является оригиналом. Никто не имел к нему доступа, кроме него самого и Северной Дакоты.

Дверцы автобуса открылись, но из него никто не вышел. Дизельный двигатель взревел, набирая обороты, и в тот момент, когда автобус уже готов был тронуться, из соседнего бара выскочили трое молодых людей. Они бежали за уже движущимся автобусом, крича и размахивая руками.

Джабба обильно полил приправой кусок пирога на тарелке. - Что-что. - Как это тебе нравится. Он аккуратно размазал приправу кончиком салфетки. - Что за отчет.