Lesson test: Random variables and expectation

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Random variables and expectation

1.[2p]
What is a random variable?

A function from the sample space to the real numbers

Correct

A number chosen at random from a range

The answer is: A function from the sample space to the real numbers

An event that may or may not occur

The answer is: A function from the sample space to the real numbers

A probability that changes between trials

The answer is: A function from the sample space to the real numbers

2.[1p]
Two fair dice are rolled. What is the expected value of the total?

CorrectNot quite: 7

3.[3p]
A European roulette wheel has $37$ pockets, $18$ of them red. What is the expected gain per unit staked on red? Give the answer to three decimal places, with its sign.

CorrectNot quite: -0.027

4.[2p]
For one roll of a fair die, what is $E [X^{2}]$ ? Give the answer to three decimal places.

CorrectNot quite: 15.167

5.[3p]
What does $E [X + Y] = E [X] + E [Y]$ require of $X$ and $Y$ ?

Nothing at all beyond being defined on the same sample space

Correct

That they be independent

The answer is: Nothing at all beyond being defined on the same sample space

That they be identically distributed

The answer is: Nothing at all beyond being defined on the same sample space

That they be non-negative

The answer is: Nothing at all beyond being defined on the same sample space

6.[3p]
Five cards are dealt from a full deck. What is the expected number of hearts?

CorrectNot quite: 1.25

7.[3p]
$n$ letters are placed at random into $n$ addressed envelopes. What is the expected number that reach the right address, for $n = 1000$ ?

CorrectNot quite: 1

8.[3p]
A fair die is rolled six times. What is the expected number of distinct faces seen? Give the answer to two decimal places.

CorrectNot quite: 3.99

9.[2p]
Two propositions with the same expected value are equally attractive.

True

The answer is: False

False

Correct

1.[2p] What is a random variable?

2.[1p] Two fair dice are rolled. What is the expected value of the total?

3.[3p] A European roulette wheel has 37 pockets, 18 of them red. What is the expected gain per unit staked on red? Give the answer to three decimal places, with its sign.

4.[2p] For one roll of a fair die, what is E[X2]? Give the answer to three decimal places.

5.[3p] What does E[X+Y]=E[X]+E[Y] require of X and Y?

6.[3p] Five cards are dealt from a full deck. What is the expected number of hearts?

7.[3p] n letters are placed at random into n addressed envelopes. What is the expected number that reach the right address, for n=1000?

8.[3p] A fair die is rolled six times. What is the expected number of distinct faces seen? Give the answer to two decimal places.

9.[2p] Two propositions with the same expected value are equally attractive.

1.[2p]
What is a random variable?

2.[1p]
Two fair dice are rolled. What is the expected value of the total?

3.[3p]
A European roulette wheel has $37$ pockets, $18$ of them red. What is the expected gain per unit staked on red? Give the answer to three decimal places, with its sign.

4.[2p]
For one roll of a fair die, what is $E [X^{2}]$ ? Give the answer to three decimal places.

5.[3p]
What does $E [X + Y] = E [X] + E [Y]$ require of $X$ and $Y$ ?

6.[3p]
Five cards are dealt from a full deck. What is the expected number of hearts?

7.[3p]
$n$ letters are placed at random into $n$ addressed envelopes. What is the expected number that reach the right address, for $n = 1000$ ?

8.[3p]
A fair die is rolled six times. What is the expected number of distinct faces seen? Give the answer to two decimal places.

9.[2p]
Two propositions with the same expected value are equally attractive.