Lesson test: Continuous random variables

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Continuous random variables

1.[3p]
Why must every single value of a continuous variable have probability zero?

Any positive common value, repeated over enough distinct values, would push the total above one

Correct

Because the density is always less than one

The answer is: Any positive common value, repeated over enough distinct values, would push the total above one

Because continuous variables are never observed exactly

The answer is: Any positive common value, repeated over enough distinct values, would push the total above one

Because the cumulative function is continuous

The answer is: Any positive common value, repeated over enough distinct values, would push the total above one

2.[2p]
A probability density function may take values greater than $1$ .

True

Correct

False

The answer is: True

3.[2p]
A variable is uniform on $[2, 8]$ . What is $P (3 \leq X \leq 5)$ ? Give the answer to four decimal places.

CorrectNot quite: 0.3333

4.[2p]
A variable is uniform on $[2, 8]$ . What is its variance?

CorrectNot quite: 3

5.[2p]
What does the cumulative distribution function $F (x)$ give?

The probability that the variable is at most $x$

Correct

The probability that the variable equals $x$

The answer is: The probability that the variable is at most $x$

The density at $x$

The answer is: The probability that the variable is at most $x$

The area under the density to the right of $x$

The answer is: The probability that the variable is at most $x$

6.[3p]
A light bulb's lifetime is exponential with mean $1000$ hours. What is the probability it lasts more than $1500$ hours? Give the answer to four decimal places.

CorrectNot quite: 0.2231

7.[3p]
Carbon-14 has a half-life of $5730$ years. What is the mean lifetime of an individual atom, in years, to the nearest year?

CorrectNot quite: 8267

8.[3p]
Which property makes the exponential the wrong model for something that wears out?

Memorylessness: the chance of failing next hour never depends on age

Correct

Its mean and standard deviation are equal

The answer is: Memorylessness: the chance of failing next hour never depends on age

Its median is shorter than its mean

The answer is: Memorylessness: the chance of failing next hour never depends on age

Its density is unbounded at zero

The answer is: Memorylessness: the chance of failing next hour never depends on age

9.[3p]
Match each law to the mechanism that produces it.

Binomial
Geometric
Poisson
Exponential
Uniform

successes in a fixed number of trials
time until the next Poisson event
trials until the first success
many trials, each very unlikely
no part of an interval favoured

Show the answer

Binomial: successes in a fixed number of trials Geometric: trials until the first success Poisson: many trials, each very unlikely Exponential: time until the next Poisson event Uniform: no part of an interval favoured

1.[3p] Why must every single value of a continuous variable have probability zero?

2.[2p] A probability density function may take values greater than 1.

3.[2p] A variable is uniform on [2,8]. What is P(3≤X≤5)? Give the answer to four decimal places.

4.[2p] A variable is uniform on [2,8]. What is its variance?

5.[2p] What does the cumulative distribution function F(x) give?

6.[3p] A light bulb's lifetime is exponential with mean 1000 hours. What is the probability it lasts more than 1500 hours? Give the answer to four decimal places.

7.[3p] Carbon-14 has a half-life of 5730 years. What is the mean lifetime of an individual atom, in years, to the nearest year?

8.[3p] Which property makes the exponential the wrong model for something that wears out?

9.[3p] Match each law to the mechanism that produces it.

1.[3p]
Why must every single value of a continuous variable have probability zero?

2.[2p]
A probability density function may take values greater than $1$ .

3.[2p]
A variable is uniform on $[2, 8]$ . What is $P (3 \leq X \leq 5)$ ? Give the answer to four decimal places.

4.[2p]
A variable is uniform on $[2, 8]$ . What is its variance?

5.[2p]
What does the cumulative distribution function $F (x)$ give?

6.[3p]
A light bulb's lifetime is exponential with mean $1000$ hours. What is the probability it lasts more than $1500$ hours? Give the answer to four decimal places.

7.[3p]
Carbon-14 has a half-life of $5730$ years. What is the mean lifetime of an individual atom, in years, to the nearest year?

8.[3p]
Which property makes the exponential the wrong model for something that wears out?

9.[3p]
Match each law to the mechanism that produces it.