Take a multiple choice test on probability distributions.
In this test you will be asked questions about the following topics:
Bernoulli distribution
Binomial distribution
Poisson distribution
Exponential distribution
When you give a right answer, it is marked with .
When you give a wrong answer, it is marked with and the right answer is marked with .
When you answer a question, your score is updated and you can read an explanation of the right answer before going to the next question.
Are you ready?
Question 1. Suppose you buy a lottery ticket. You can either win dollar (with probability ) or win nothing (with probability ). The amount you win is a random variable and it has a:
Gamma distribution
normal distribution
exponential distribution
Bernoulli distribution
Explanation. Denote by the amount you win. can take only two values ( and ), with probabilities and .This corresponds to the definition of a Bernoulli random variable. For more details, see the lecture entitled Bernoulli distribution.
Question 2. Consider a Bernoulli random variable , with support and probability mass function:The variance of is:
Explanation. The variance of a Bernoulli random variable is calculated as follows:For more details, see the lecture entitled Bernoulli distribution.
Question 3. Consider a Bernoulli random variable , with support and probability mass function:The moment generating function of is:
Explanation. The moment generating function of a Bernoulli random variable is calculated as follows:For more details, see the lecture entitled Bernoulli distribution.
Question 4. Suppose you independently flip a coin times and the outcome of each toss can be either head (with probability ) or tails (also with probability ). Denote by the number of times the outcome is tails (out of the tosses). The random variable has a:
Bernoulli distribution
Poisson distribution
binomial distribution
exponential distribution
Explanation. is the sum of independent Bernoulli random variables (that take value in case the outcome is tails and in case the outcome is head). As we explained in the lecture entitled binomial distribution, the sum of independent Bernoulli random variables is a binomial random variable.
Question 5. Suppose you independently play a game times. Each time you play, the probability of winning is . Denote by the number of games you win (out of the games you play). Denote by the probability mass function of . Then, when :
Explanation. is the sum of independent Bernoulli random variables that take value in case you win the game (with probability ) and in case you lose (with probability ). As we explained in the lecture entitled binomial distribution, the sum of independent Bernoulli random variables is a binomial random variable. The probability mass function of a binomial random variable is:Therefore, in our case, when :See the lecture entitled binomial distribution for further details.
Question 6. Suppose you independently throw a dart times. Each time you throw a dart, the probability of hitting the target is . Denote by the number of times you hit the target (out of the total times you throw a dart). The expected number of hits is:
Explanation. is the sum of independent Bernoulli random variables that take value in case you hit the target (with probability ) and in case you miss it (with probability ). As we explained in the lecture entitled binomial distribution, the sum of independent Bernoulli random variables is a binomial random variable. The expected value of a binomial random variable is:See the lecture entitled binomial distribution for further details.
Question 7. Suppose you independently throw a dart times. Each time you throw a dart, the probability of missing the target is . Denote by the number of times you hit the target (out of the total times you throw a dart). The variance of is:
Explanation. is the sum of independent Bernoulli random variables that take value in case you hit the target (with probability ) and in case you miss it (with probability ). As we explained in the lecture entitled binomial distribution, the sum of independent Bernoulli random variables is a binomial random variable. The variance of a binomial random variable is:See the lecture entitled binomial distribution for further details.
Question 8. The probability that a working steel cutting machine breaks down during a time interval (given that it has not broken down before that time interval) is approximately proportional to the length of that time interval. Denote by the total time elapsed before the machine breaks down. can be assumed to have a:
Gamma distribution
Poisson distribution
Exponential distribution
Uniform distribution
Explanation. Denote by the time we need to wait before a certain event occurs. Suppose is unknown (it is a random variable). Take any time interval . If the probabilityis (approximately) proportional to , then has an exponential distribution. In other words, has an exponential distribution if the conditional probability that the event occurs during a time interval is proportional to the length of that time interval. See the lecture entitled Exponential distribution for more details on the exponential distribution.
Question 9. Let be an exponential random variable with parameter . The distribution function of is:
Explanation. The probability density function of an exponential random variable is:The distribution function is the integral of the probability density function:See the lecture entitled Exponential distribution for more details on the distribution function of the exponential distribution.
Question 10. The probability that a new customer enters our shop during a given minute is approximately , irrespective of how many customers have entered the shop during the previous minutes. Assume that the total time we need to wait before a new customer enters our shop (denote it by ) has an exponential distribution. Then, the probability that no customer enters the shop during the next hour is:
Explanation. Time is measured in minutes. Therefore, the probability that no customer enters the shop during the next hour is:where is the distribution function of . Since is an exponential random variable with rate parameter , its distribution function is:Therefore:See the lecture entitled Exponential distribution for more details.
Question 11. Denote by the time (in years) that will elapse before a certain firm goes bankrupt. has an exponential distribution and its expected value is years. The probability density function of is:
Explanation. The probability density function of an exponential random variable with rate parameter is:and its expected value is:In this case the expected value is 10 years. Therefore:which implies:Thus, the probability density function of is:See the lecture entitled Exponential distribution for more details on the exponential distribution and its expected value.
Question 12. The time between the arrival of one customer and the arrival of the next customer has an exponential distribution and is independent of previous arrivals. The number of customers that will arrive during the next hour has:
a Poisson distribution
a Gamma distribution
an exponential distribution
a binomial distribution
Explanation. If a certain event can occur many times within a unit of time and the time elapsed between two successive occurrences of the event has an exponential distribution (independent of previous occurrences), then the total number of occurrences within a unit of time has a Poisson distribution. See the lecture entitled Poisson distribution for more details on the Poisson distribution.
Question 13. Let be a Poisson random variable with rate parameter . Then:
Explanation. For , the probability mass function of a Poisson random variable is:Therefore, for :See the lecture entitled Poisson distribution for more details on the Poisson distribution.
Question 14. The time between the arrival of one customer and the arrival of the next customer has an exponential distribution with parameter and is independent of previous arrivals. Time is expressed in hours. Denote by the number of customers that will arrive during the next hour. The expected value of is:
Explanation. If a certain event can occur many times within a unit of time and the time elapsed between two successive occurrences of the event has an exponential distribution with parameter (independent of previous occurrences), then the total number of occurrences within a unit of time has a Poisson distribution with parameter . Therefore has a Poisson distribution with parameter . The expected value of a Poisson random variable with parameter is:See the lecture entitled Poisson distribution for more details on the Poisson distribution.
Question 15. The sum of two independent exponential random variables with common parameter has:
an exponential distribution
a binomial distribution
a Gamma distribution
a Poisson distribution
Explanation. The sum of two independent exponential random variables with common parameter has a Gamma distribution. See the lecture entitled Exponential distribution for a proof of this fact.
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