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Probability Theory Trivia Quiz

Have you ever wondered how much you know about probability, the mathematical treatment of randomness of life? Take this quiz to find out!

A multiple-choice quiz by irving9918. Estimated time: 6 mins.

Author
irving9918

Time
6 mins

Type
Multiple Choice

Quiz #
350,695

Updated
Dec 03 21

# Qns
10

Difficulty
Difficult

Avg Score
5 / 10

Plays
289

One at a Time - Single Page - Timed Game

Question 1 of 10

1. Combinatorics: Assume the balls are indistinguishable in all aspects except color. In an urn with 5 black balls and 3 white balls, how many ways can you pick 2 black balls and 1 white ball at once? Hint

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Question 2 of 10

2. Definition of Probability: Which of the following statements about probability is INCORRECT? Hint

If A and B are two disjoint events, then P(A and B) = P(A)*P(B)

If A and B are two disjoint events, then P(A or B) = P(A)+P(B)

If S is the sample space, then P(S) = 1

If A is an event, then P(not A) = 1-P(A)

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Question 3 of 10

3. Conditional Probability: There are four closed doors. Behind one of them is a car, and the other three are goats. You pick one door at random. The game host then opens two doors you didn't select that have goats behind each one. Now there are two doors unopened. The host then asks you whether you want to switch to the other unopened door. What is your probability of winning the car if you switch? Hint

1/4

2/3

1/2

3/4

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Question 4 of 10

4. Discrete Random Variables: If X~Binomial(10,0.4), which of the following is true about the mean and variance of X? Hint

E(X) = 4, Var(X) = 1.6

E(X) = 4, Var(X) = 2.4

E(X) = 4, Var(X) = 3.6

E(X) = 6, Var(X) = 3.6

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Question 5 of 10

5. Continuous Random Variables: If U~Uniform(0,1) and X~Exponential(1). Then Hint

X = -log U

X = exp(U)

X = log U

X = -exp(U)

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Question 6 of 10

6. Joint Random Variables: Let f(x,y) be the joint probability mass function (PMF) or probability density function (PDF) of random variables X and Y. Let f(x) and f(y) be their marginal PMF/PDF, respectively. Then which of the following is true when X and Y are independent? Hint

f(x,y) = f(x)*f(y)

f(x,y) = f(x) or f(x,y) = f(y)

f(x,y) = f(x)+f(y)

f(x,y) = f(x)/f(y)

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Question 7 of 10

7. Conditional Expectation: If X|Y~Normal(2Y,3Y) and Y~Poisson(5). Find E(X). Hint

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Question 8 of 10

8. Central Limit Theorem: Let X1,...,Xn be independent and identically distributed with Poisson(m). When n is large, what is the approximate distribution of ∑Xi, i=1,...,n? Hint

Normal(mn,mn^2)

Normal(mn,m)

Normal(m,mn)

Normal(mn,mn)

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Question 9 of 10

9. Generating Functions: Which of the following statements is INCORRECT about the moment generating function (MGF)? For reference, MGFx(t) = E[exp(tX)]. Hint

The MGF is a positive function

The MGF does not always exist for every distribution

When every moments of X exists, then the MGF of X exists

The MGF uniquely determines the distribution

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Question 10 of 10

10. Order Statistics: The minimum of n independently identically distributed (i.i.d.) Exponential(a) random variables has what distribution? Hint

Exponential(a/n)

Gamma(n,a)

Exponential(n/a)

Exponential(an)

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Quiz Answer Key and Fun Facts

Answer: 30

There are (5 choose 2) = 5*4/2 = 10 ways to pick 2 black balls.
There are 3 ways to pick 1 white ball.
So there are a total of 10*3 = 30 ways to pick 2 black balls and 1 white ball.

2. Definition of Probability: Which of the following statements about probability is INCORRECT?

Answer: If A and B are two disjoint events, then P(A and B) = P(A)*P(B)

If A and B are two disjoint events, then P(A and B) = P(A)*P(B) is incorrect. In order for the equation to hold, A and B must be independent, not disjoint. All other answers are definitions of probability.

Answer: 3/4

This is the modified version of the Monty Hall problem. Consider the following two cases:
(1) With probability 1/4, the door you originally picked has a car behind it:
In this case, the other three doors have goats behind each one. The host can then randomly choose two of the three other doors to open, and you will get a goat by making the switch.
(2) With probability 3/4, the door you originally picked has a goat behind it:
In this case, two of the other three doors have goats behind each one and one of the other three doors have a car behind it. The host is forced to open the two doors with goats behind them. So you will win the car by making the switch.
So the probability you will win the car by making the switch is 3/4.

4. Discrete Random Variables: If X~Binomial(10,0.4), which of the following is true about the mean and variance of X?

Answer: E(X) = 4, Var(X) = 2.4

For Binomial(n,p) distribution, the mean is np, and the variance is np(1-p). Here, n = 10 and p = 0.4. So the mean is 10*0.4 = 4, and the variance is 10*0.4*(1-0.4) = 2.4.

5. Continuous Random Variables: If U~Uniform(0,1) and X~Exponential(1). Then

Answer: X = -log U

The exponential distribution can be generated by taking the negative of the logarithm of the uniform distribution.

Answer: f(x,y) = f(x)*f(y)

Independence always leads to factorization of joint PMF/PDF into respective marginals. Do not confuse joint PMF/PDF with conditional PMF/PDF here. It will be correct to say f(x|y) = f(x) when X and Y are independent, but not f(x,y) = f(x).

7. Conditional Expectation: If X|Y~Normal(2Y,3Y) and Y~Poisson(5). Find E(X).

Answer: 10

Use the law of iterated expectations: E(X) = E[E(X|Y)] = E(2Y) = 2*E(Y) = 2*5 = 10.

8. Central Limit Theorem: Let X1,...,Xn be independent and identically distributed with Poisson(m). When n is large, what is the approximate distribution of ∑Xi, i=1,...,n?

Answer: Normal(mn,mn)

The central limit theorem says that (∑Xi-�n)*n^(-1/2) converges to N(0,�). So ∑Xi is approximately N(�n,�n).
You can also see this by computing the mean and variance of ∑Xi:
E(∑Xi) = ∑E(Xi) = ∑� = �n
Var(∑Xi) = ∑Var(Xi) = ∑� = �n where the first equality follows from the independence of Xi.

9. Generating Functions: Which of the following statements is INCORRECT about the moment generating function (MGF)? For reference, MGFx(t) = E[exp(tX)].

Answer: When every moments of X exists, then the MGF of X exists

The MGF of the LogNormal distribution does not exist even though all moments exists. To compute the moments of the LogNormal distribution, we find it convenient to use the MGF of the normal distribution. For example, when X~LogNormal(�,V), then Y=log X~Normal(�,V), and thus
E(X^n) = E[exp(nY)] = MGFy(n) = exp(�n+0.5*V*n^2)
However, MGFx(t) = E[exp(tX)] = E{exp[t*exp(Y)]} = ∫exp[t*exp(y)]*f(y)dy blows up, where f(y) is the density of Normal(�,V).

The MGF is a positive function because exp is a positive function.

10. Order Statistics: The minimum of n independently identically distributed (i.i.d.) Exponential(a) random variables has what distribution?

Answer: Exponential(an)

Let X1,...,Xn be n i.i.d. Exponential(a). Then
P(Xj>x) = exp(-ax)
P[min(X1,...,Xn)>x] = P(X1>x,...,Xn>x) = P(X1>x)...P(Xn>x) = exp(-anx)
So min(X1,...,Xn)~Exponential(an)

Source: Author irving9918

This quiz was reviewed by FunTrivia editor WesleyCrusher before going online.
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