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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

Reservations for a theater performance are made according to a Poisson process with rate ???? = 5 reservations per hour. If there are currently 20 seats available, what is the probability that all
Screws produced by a machine in a large company are packaged in packs of 50 screws each. The company believes that a pack cannot be sold if it contains at least three defective screws. If it is known
A bus traveling between Paris and Lyon in France is about to depart in five minutes.The bus has 52 seats and they have all been booked. Anne and Sophie have just arrived at the bus station without a
The distribution of X is(a) hypergeometric (b) binomial (c) Poisson(d) negative binomial (e) none of the above
If X has a (????) distribution and we know that the standard deviation of X is three times higher than its mean, then the value of ???? is(a) 1∕9 (b)1∕3 (c) 1 (d) 3 (e) 9
The number of cars arriving at a parking station follows a Poisson process{N(t) ∶ t ≥ 0} with rate ???? cars for a five-minute period. Suppose that the variance of the number of cars to arrive in
Then the value of ???? is(a) 18 (b) 9∕2 (c)9∕10 (d) 9∕4 (e)3∕2
The number of cars arriving at a gas station follows a Poisson process {N(t) ∶t ≥ 0} with rate ???? = 7 cars for a five-minute period. The expected number of cars to arrive between 9:00 a.m. and
Assume that X has a b(n, p) distribution. If it is known that the mean of X is 6 and its standard deviation is 2, then the values of n and p are(a) n = 18, p = 2∕3 (b)n = 9, p = 2∕3 (c)n = 18, p
Susan throws three dice and Andrew also throws three dice. What is the probability that they get the same number of sixes?
The probability of error in the transmission of a binary digit over a communication channel is 2 ⋅ 10−4. If 5000 digits are transmitted,(i) find the probability that at least two errors
The price of a certain stock either increases during a trading day by $1 and this happens with probability 0.7 or decreases by $1. What is the probability that, after 20 trading days, the price of
We ask five persons which day of the week they were born. Find the probability that(i) no one was born on a Tuesday;(ii) at least three persons were born during a weekend;(iii) not all five were born
From an urn that contains 60 balls numbered 1–60, we select 3 balls with replacement. Let X be the random variable which denotes the number of balls selected, having a number which is a multiple of
Nicky, who is a wine taster, tries five California wines and six Bordeaux wines and she is asked to assign a number to each wine, according to her preference, so that the best wine receives number
Moreover, if the number of breakdowns equals x, the financial damage incurred due to the repair costs and the fact that the machine is not working equals ax2 + bx, where a and b are known positive
A scientific institute has n1 male and n2 female members. In the last elections of the institute, n members did not vote (n < n1 + n2). If all institute members have the same probability of
Anew law proposal is submitted to the parliament of a country. There are n members of the government party, who will all certainly vote in favor of the proposal, m members of the leading opposition
While information is transmitted to a computer system, a so-called parity check takes place as follows: suppose that the system uses words with n binary digits(bits), i.e. each digit is either 0 or
Let X be a variable that follows the Poisson distribution with parameter ???? >
If r is an integer with r ≥ 2, define the random variable (i) Prove that r-1 Y = (x + 1) = (x + 1)(x+2) (X+r 1). El E (+) = - = (1-20-14). (ii) Calculate the expectation (1+*)3 and compare this
Let Xn be the number of successes in n independent Bernoulli trials, each with a success probability p, and(i) Using the law of total probability and the partition {B1, B2} with B1 = {X1 = 0}, and B2
In a sequence of n + k independent Bernoulli trials with success probability p in each trial, let X be the number of successes in the first n trials and Y be the total number of successes out of the
The probability that a car is stolen when it is parked overnight in an unsafe area of a city is 10%. If there are 12 cars parked on a particular street of that area, what is the probability that
The number of animals caught in an animal trap during an hour has the Poisson distribution with parameter ????. If it is known that the probability that no animal is trapped during an hour equals
An electronic game machine makes a profit of c dollars per hour of service. The machine, however, breaks down at random times. In particular, it is assumed that the number of breakdowns follows a
A bowl contains a blue and b yellow chips. From the bowl, we select successively and without replacement, until the nth blue chip is selected. Let X be the number of chips selected until this
The number of phone calls that Nicky receives on her mobile phone follows a Poisson process with a rate ???? = 3 calls per hour.(i) Find the probability that she will receive three calls between 6:00
An employee in a telephone sales company estimates that she has a 15% chance of closing a sale after each phone call.(i) What is the expected number of sales she will make after 40 calls?(ii) What is
George and Faye enter a quiz in which they have to answer independently 10 “True or False” questions.(i) Assuming that they are both purely guessing, so that both have a probability of 1∕2 of
Let X be a random variable with the negative binomial distribution with parameters r and p.(i) Let Y = X − r. Find the probability function, the expectation, and variance of Y.(ii) Assume that, as
When Lucy goes shopping, the number of items she buys follows a Poisson distribution with rate ????1 =
However, if her friend Irene, who is a shopaholic, goes with her, then she tends to buy more items, so that the number of purchased items still has a Poisson distribution but with a new rate ????2 =
In Exercise 14 of Section 5.1 and in Exercise 12 of Section 5.4, you were asked to show that the probability function of both the binomial and the Poisson distribution satisfies the recursive
Let X have a distribution on the nonnegative integers {0, 1, 2,…} such that (5.18)holds.(i) Assuming that E(X) exists, show that(ii) If we assume that E[X(X − 1)] exists, then it is given byFrom
Customers arrive at a bookstore with a rate of ???? = 20 customers per hour. It is estimated that 30% of the customers buy a particular book which is this month’s best-seller. What is the
From an urn, which contains a white and b black balls,we select n balls with replacement.However, each time a ball is put back into the urn, we also put s more balls of the same color into the urn.
Then, next to each word which is transmitted, another parity digit is placed such that the total number of 1’s representing a word is an even integer. When the word with the n + 1 digits is
Alice wants to buy a new iPad, and she would like to try her chances with the lottery for the first time. After a little research, she finds out that a single ticket of the lottery (i.e. if you
At the NBA playoff finals, the championship is awarded in a best-out-of-seven series of games, i.e. the first team to reach four wins is the champion. Assume that in each game, the two teams, say A
Emily is in the final semester of her studies at the University. She has completed successfully all compulsory courses needed for her degree and shemust pass at least four exams to finish her studies
Let X be a random variable that has the geometric distribution with parameter p(0 < p < 1), and let f be the probability function of X.(i) Verify that f satisfies the recursive relationship f (x) =
Rodney, who sells newspapers on the street, buys each newspaper for 35 cents and he sells it for 50 cents. However, he cannot return any unsold newspapers. He has estimated that the number of
In a number generated in this way, what is the expected number of decimal digits(i) before the digit 5 appears for the first time?(ii) before one of the digits 2, 3, 4 appears for the first
A computer selects decimal digits of a random number in the interval (0, 1) with equal probability among the digits 0, 1, 2,…,
In a match of women’s tennis, the winner is the player who first wins two sets.Suppose two players, Martina and Steffi, play against each other and that the probability that Martina wins any
Peter plays the following game: he rolls a die successively and he receives k2 dollars if a five appears for the first time in the kth row. Find Peter’s expected turnover from this game.
The probability that someone tests positive in a medical examination for a disease is 6%. What is the probability that(i) the 8th person who is examined is the first who tests positive?(ii) the 12th
Danai and Bill each have a die and throw it simultaneously. If both dice land on an even number, the game ends. If not, they continue with another round of each throwing their die.(i) What is the
The probability that a digital signal is transmitted incorrectly is p. Suppose a particular signal is transmitted repeatedly and that successive transmissions occur every two minutes. What is the
If we throw a fair die successively, find the probability that a four appears for the third time(i) at the 10th throw;(ii) after the first 20 throws of the die.
Let X be a random variable that follows the binomial distribution with parameters n and p, and denote by X∕n the proportion of successes in the n trials. Show that for any positive real number ????
Let X be a random variable with a b(n, p) distribution. Show that for any positive integer r, the rth factorial moment of X????(r) = E[(X)r] = E[X(X − 1)(X − 2) · · · (X − r + 1)]is given by
A basketball player makes free throws successfully with a probability 0.85. In a particular game, he attempted 16 free throws and scored in 11 of them.(i) What is the probability that the five throws
A jury has three members. Each of these members makes the correct decision with probability p, independent of the other jury members. The jury’s decision is based on the majority rule.(i) What is
We throw a die repeatedly and we stop when the outcome is either five or six.(i) What is the probability that k throws will be needed and the outcome of the kth throw is six?(ii) What is the
In a series of Bernoulli trials with success probability p, find the probability that k successes in a row occur prior to n failures in a row.(Hint: Let  be the event of interest, namely,  ∶
The maximum number of independent Bernoulli trials in an experiment is 3r, and the success probability in each trial is p. Trials terminate when either• r successes have occurred, or • it is not
In a lottery played in several countries, players have to choose 6 numbers from 1 to
A box contains six white stones and six black stones. Peter selects three stones at random without replacement. Let Y be the random variable that represents the number of white stones minus the
In a school class with 20 male students, 7 of them are white. The sports teacher wants to select five students for the basketball team (regular team) and seven more students as substitutes in that
We select cards from a pack of 52 cards and let X be the number of cards picked until the first queen is drawn. Find the probability function and the expected value of X.
An urn contains a red balls and b black balls. We select n balls from this urn with replacement. If X denotes the number of red balls in the sample, derive the probability function and the expected
A matchbox contains normally 40 matches. We select three matchboxes at random and we find seven matches in total to be defective. What is the probability that there are at least two defective matches
From a usual pack of 52 cards, we select 3 cards at random without replacement.Find the probability that(i) no ace is selected;(ii) a queen is selected;(iii) at least one queen is selected;(iv) no
There are 10 blue and 16 red chips in a bowl. If we select five chips at random, what is the probability that more red chips are drawn than blue ones?
During an excavation, an archaeologist has discovered bones of 12 animals; 5 of these bones belong to rhinoceros, 4 belong to a mammoth, and 3 to a certain type of hippopotamus. She wants to select
An urn contains 15 balls numbered 1–15. We select three balls without replacement.Find the probability that(i) at least one number drawn is a prime;(ii) exactly one number drawn is a prime.
Assume that a lake contains a total of N fish. Suppose we catch r among these fish, mark them in some way, and then put them back in the lake. After a certain amount of time, so that the marked fish
16 players enter a table tennis tournament. Three of the players who enter the tournament are left handers while the rest are right handers. Assuming that initially all players are equally likely to
On a supermarket shelf, there are 45 packs of cereals. Among these, there are five packs whose sell-by date is less than a week from now. Lena selects four packs of cereals at random and intends to
Let X be a random variable that follows the Nb(r, p) distribution. Show that for any positive integer k, the quantityIncidentally, the quantity ????[k] is called an ascending factorial moment of
Anna and Steve play the following game. A die is thrown until either a five or a six appears for the first time. If the number of throws X is even, then Anna gives Steve the amount of a dollars; if X
Solve the Banach matchbox problem (Example 5.14) if the two matchboxes do not contain initially the same number of matches. In other words, assume that the box in the right-hand pocket has N1 matches
At the end of the academic year, Sophie will sit in six exams, of which three she considers “easy” as she feels that she has a probability p of passing the exam.Sophie thinks the other three
Lisa, Tony, and Tom take part in a TV quiz and at some point they have to answer the same question. They do not have the same probability of answering the question correctly, but their respective
Prove the following, which is a more general form of Markov’s inequality than the one given in Proposition 4.11. Let X be a random variable that takes nonnegative values and for which E(Xk) exists
The probability function of a discrete random variable X is given byfor some integer n ≥ 2 and a suitable constant c.(i) Prove that c = 2.(ii) Find the expected value E(X) and the variance
A coin is tossed until either a head or four tails have occurred, and let X stand for the total number of coin tosses in this experiment.(i) What is the range of values for X?(ii) Calculate the
An electronic system consists of two parts that function independently of one another. The lifetime T (in hours) for each part of the system is a random variable with distribution function(i) Find
The weekly demand for a magazine in a news shop can be represented by a random variable with the following probability function:The owner of the shop buys each copy of the magazine for $4 and she
Simon and Paul place n balls numbered 1, 2,…, n in an urn and agree to play the following game. Simon pays the amount of a dollars to Paul, he then selects k balls randomly and receives from Paul
A small internet company sells ebooks. The company has paid $200 to obtain a new book electronically and sells each copy of the book at a price of $30. Let X stand for the number of sales the company
A random variable X has its distribution function as(i) What is the range of values for X?(ii) Calculate the probabilities P(X = 2), P(X = 2|X (iii) Obtain the expected value of the variables X, X2
Let X be the largest outcome in the throws of two dice. In Example 4.8, we found that the probability function of X is given by(i) Write down the distribution function of X.(ii) Calculate the
A discrete random variable X can take on only four values a1, a2, a3, a4 with respective probabilities(i) What are the admissible values for the parameter ?????(ii) For ai = i, find the value of ????
If it is known that p ≠ 1∕2 and that the mean and the variance of X are E(X) = 3 and Var(X) = 1, find the values of a and p.
A random variable X takes only the values a (with probability p) and 2a (with probability 1 − p) for some 0 < p
Let X be a discrete random variable with probability function(i) Show that the variable Y = |X| has probability functioni.e. Y has the discrete uniform distribution on the set {0, 1, 2, 3,…, n −
A random variable X has range RX = {0, 1, 2,…}, and suppose thatThen the expected value of X equals (a) 1∕3 (b) 3 (c) 3∕2 (d)2∕3 (e)1∕2 P(X > k): k= 0,1,2,...
For the random variable X, we know that P(X = 1) = 1∕2, P(X = 2) = 1∕4, P(X = 4) = 3∕16, P(X = 8) = 1∕16.Then an upper bound for the probability P(X ≥ 4) from Markov’s inequality is(a)
For the random variable X, it is known that E(X4) = 50, E(2X2 + 7) = 17.Then the variance of the variable Y = X2 equals(a) 25 (b) 50 (c) 75 (d) 94 (e) 100
The probability function of a variable X is given by 4(x-1) f(x)= 3 x Ry = {2, 3, 4, ...}. (i) Show that 2n+1 P(X > n) = n = 2,3,... 3" (ii) Verify that the variable X satisfies the inequality
Let X be a discrete random variable and g be a function such that the expectation E[g(X)] exists. Show that for the expectation of the random variable |g(X)|, we have|E[g(X)]| ≤ E[|g(X)|].
A machine produces metal discs whose diameter D is a random variable with distribution functionwhere a 0). 0, t-a F(t)= b-a 1. ta, a b
Find the investor’s expected profit.
An investor invests the same amount of money in three funds. In each of these, he estimates that there is a probability p = 3∕5 of making a profit of $15 000, or else he loses $10
Let X be a random variable having the binomial distribution with parameters n and p.(i) Show that its probability function f (x), x = 0, 1,…, n, can be calculated via the recursive relationwith
Among the claims that are received by an insurance company, it is estimated that 3% exceed a specified amount c and those are registered as “large claims.”(i) What is the probability that in the
Let X be a random variable having the binomial distribution with parameters n and p for some p ∈ (0, 1) and Y be another random variable having the binomial distribution with parameters n and q,

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