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# System prepended metadata

title: Prob2025 Exam

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# Prob2025 All Exams

@ntu-csie prob_exam (113-2)

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# Prob2025 Small Quiz 3/20

## P1 solution count
How many distinct solutions does the following equation have?

$X1 + X2 + X3 + X4 = 100$,
such that $x_1 ∈ {1,2,3,...}, x_2 ∈ {2,3,4,...}, X_3, X_4 ∈ {0,1,2,3,...}$

## P2 Liar's Bar
Liar's Bar is a bluffing poker game for four players using a 20-card deck consisting of six Aces, six Kings, six Queens, and two Jokers. One card is drawn to be the "Lord" (except Joker), which is used for the round's bets, before the shuffle. Each player is dealt five cards afterward. Players take turns declaring 1-3 "Blinds" (Cards placed face down) of the same rank (Q, K, A) as the Lord. The next player can challenge the previous player's declaration, and if the declared Blinds are a lie, the liar loses; if not, the challenger loses. Jokers are treated as Lord, and if a player fails to challenge a Joker, all players except the declarer will lose.

(a) Sakura Miko, a legendary bettor, is playing Liar's Bar to strike it rich. In a game with Q as the Lord. She holds Q, K, A, and Joker at hand. A rival declares two Queens now, what is the probability that the rival actually doesn't have more than two Queens or Jokers? (Without Combination of Q+Q or Q+Joker)

(b) Following (a), Miko has 90% confidence that her challenge will succeed if the opponent is bluffing (has less than 2 Qs or Jokers) and 15% confidence if the opponent is not bluffing. Given that Miko's challenge was successful, determine the probability that the opponent is bluffing.

## P3 Liar's Bar
Let f(x) be the probability mass function (pmf) of a discrete random Variable X.

   $f(x)=P(X=x)=\frac{c}{(x+5)(x+6)},x=0,1,2,...$

   (a) Solve c.
   (b) Compute $P(X \leq k)$.

## P4 MTTF
Mean Time to Failure (MTTF) is a critical parameter in system design, which can be calculated as the mathematical expectation of the time that the first failure occurs. Suppose there is a system consisting of component A and B. The table below shows the failure probabilities of component A and B at each time point.

For simplicity, we assume that both components are guaranteed to fail if the time reaches t = 3. The failure probabilities at different time points are mutually independent, and the failure probabilities of component A and B are also independent of each other. Answer the following questions.

| Failure Probability at time t | A     | B     |
| :----------------------------- | :---- | :---- |
| t = 1                          | 40%   | 50%   |
| t = 2                          | 50%   | 60%   |
| t = 3                          | 100%  | 100%  |

(a) Compute MTTF of component A
(b) Compute MTTF of component B
\(c) Compute MTTF of the system if A and B are connected in series (The system fails if either A or B fails)
(d) Compute MTTF of the system if A and B are connected in parallel (The system fails if both A and B fail)

## P5 disease testing
In group testing for a certain disease, a blood sample was taken from each of n individuals and part of each sample was placed in a common pool. The latter was then tested. If the result was negative, there was no more testing and all individuals were declared negative with one test. If, however, the combined result was found positive, all individuals were tested, requiring n + 1 tests. If p = 0.05 is the probability of a person's having the disease and n = 5, compute the expected number of tests needed, assuming independence.

|  | p= | 0.01   | 0.02   | 0.03   | 0.04   | 0.05   | 0.06   | 0.07   | 0.08   | 0.09   | 0.1    |
|---|---|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| n=2 | x=0 | 0.9801 | 0.9604 | 0.9409 | 0.9216 | 0.9025 | 0.8836 | 0.8649 | 0.8464 | 0.8281 | 0.8100 |
|  | 1 | 0.9999 | 0.9996 | 0.9991 | 0.9984 | 0.9975 | 0.9964 | 0.9951 | 0.9936 | 0.9919 | 0.9900 |
|  | 2 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 3 | 0 | 0.9703 | 0.9412 | 0.9127 | 0.8847 | 0.8574 | 0.8306 | 0.8044 | 0.7787 | 0.7536 | 0.7290 |
|  | 1 | 0.9997 | 0.9988 | 0.9974 | 0.9953 | 0.9928 | 0.9896 | 0.9860 | 0.9818 | 0.9772 | 0.9720 |
|  | 2 | 1.0000 | 1.0000 | 1.0000 | 0.9999 | 0.9999 | 0.9998 | 0.9997 | 0.9995 | 0.9993 | 0.9990 |
|  | 3 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 4 | 0 | 0.9606 | 0.9224 | 0.8853 | 0.8493 | 0.8145 | 0.7807 | 0.7481 | 0.7164 | 0.6857 | 0.6561 |
|  | 1 | 0.9994 | 0.9977 | 0.9948 | 0.9909 | 0.9860 | 0.9801 | 0.9733 | 0.9656 | 0.9570 | 0.9477 |
|  | 2 | 1.0000 | 1.0000 | 0.9999 | 0.9998 | 0.9996 | 0.9992 | 0.9987 | 0.9981 | 0.9973 | 0.9963 |
|  | 3 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9999 | 0.9999 |
|  | 4 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 5 | 0 | 0.9510 | 0.9039 | 0.8587 | 0.8154 | 0.7738 | 0.7339 | 0.6957 | 0.6591 | 0.6240 | 0.5905 |
|  | 1 | 0.9990 | 0.9962 | 0.9915 | 0.9852 | 0.9774 | 0.9681 | 0.9575 | 0.9456 | 0.9328 | 0.9185 |
|  | 2 | 1.0000 | 0.9999 | 0.9997 | 0.9994 | 0.9988 | 0.9980 | 0.9969 | 0.9955 | 0.9937 | 0.9914 |
|  | 3 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9999 | 0.9999 | 0.9998 | 0.9997 | 0.9995 |
|  | 4 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
|  | 5 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

# Prob2025 Midterm 4/17

## P7 Maximum of Uniform Variables

Let $X_i \sim \text{Uniform}[0, 1]$ be mutually independent for $i = 0, 1, \ldots, n$. Let $X' = \max\{X_0, X_1, \ldots, X_n\}$. What is the probability density function (PDF) of $X'$?

> Solution: $$
\begin{aligned}
CDF_{X'}: F_{X'}(x) &= P(X' \leq x) \\
&= P((X_0 \leq x) \cap (X_1 \leq x) \cap \ldots \cap (X_n \leq x)) \\
&= \left[P(X_0 \leq x)\right]^{n+1} \\
&= x^{n+1}, \quad \text{for } 0 \leq x \leq 1 \\
PDF_{X'}: f_{X'}(x) &= \frac{d}{dx}F_{X'}(x) \\
&= \frac{d}{dx}x^{n+1} \\
&= (n+1)x^n, \quad \text{for } 0 \leq x \leq 1
\end{aligned}
$$
$$
\therefore \boxed{f_{X'}(x) =
\begin{cases}
(n+1)x^n & \text{if } 0 \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}}
$$

# Prob2025 Final 6/5

## P1 Minty Probabilities
A candy maker produces mints, and the distribution of the weights is $N(21, 0.16)$.
(a) Let $X$ equal the weight of a single mint selected at random. Find $P(20.4 < X < 21.6)$.
(b) Suppose that 6 mints are selected independently and weighed. Let $Y$ equal the number of these mints that weigh more than 20.52 grams. Find $P(Y ≥ 5)$.  
(You may refer to the standard normal table provided at the end of the question paper.)

## P2 T-Test Territory
Let the distribution of $T$ be $t(17)$.
(a) Find $t_{0.01}(17)$.  
(b) Find $t_{0.95}(17)$.  
\(c) Compute $P(−1.740 ≤ T ≤ 1.740)$.  
(You may refer to the student’s $T$ table provided at the end of the question paper.)

## P3 Cauchy Alchemy
Let $U ∼ Uniform(0, 1)$. You are asked to construct a transformation of $U$ that yields a random variable with a Cauchy distribution. Provide the explicit form of the transformation and justify why the resulting variable follows the Cauchy probability density function.

## P4 Joint PDF Jungle
Let $$f(x, y) = 1/8$ for $0 ≤ x ≤ 4$, and $y ≤ x ≤ y + 2$, be the joint PDF of $X$ and $Y$.
(a) Find $f_X(x)$, the marginal PDF of $X$.  
(b) Find $f_Y(y)$, the marginal PDF of $Y$.  
\(c) Determine $f(x | y)$, the conditional PDF of $X$ given that $Y = y$.  
(d) Determine $g(x | y)$, the conditional PDF of $X$ given that $Y = y$.  
(e) Compute $E[Y | x]$, the conditional mean of $Y$ given that $X = x$.  
(f) Compute $E[X | y]$, the conditional mean of $X$ given that $Y = y$.

## P5 Bluff Queen Miko
The Legendary bettor, Sakura Miko comes back. To win back the money she lost last time, Miko attends another Bluffing game again. Each player could play 1–3 Blinds (Cards placed face down) and Bluff in this poker game. We know that Miko has a prior probability of 50% to lie and follows an aggressive strategy. Specifically, if she plays 1 card, the probability of her lying is twice as high as if she plays 2 cards, and three times as high as if she plays 3 cards. Given her Play tendencies shown in the table below, determine the probability for each type of plays if Miko is lying:
(a) $\text{P(1 card | lying)}$
(b) $\text{P(2 cards | lying)}$ 
\(c) $\text{P(3 cards | lying)}$

| Plays tendency | Ratio |
|       ---      |  ---  |
| 1 card         |  70%  |
| 2 cards        |  25%  |
| 3 cards        |  5%   |

## P6 Exponential Trio Showdown
Let $X, Y, Z$ be independent and identically distributed exponential random variables, each with parameter $λ = 1$.  
(a) Compute the expectation of min{$X, Y, Z$}.  
(b) Compute the variance of min{$X, Y, Z$}.

## P7 Poisson Patch Probability
Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter α, where the expected number of trees per 100 m² is $α = 3$. Today, we conduct a random investigation that follows another Poisson process with parameter $λ = 3$. That is, we investigate $X ∼ Poisson(3)$ independent land patches (each 100 m² in area). What is the expected probability that **all** investigated patches contain **more than 2 trees**?

## P8 Sum of Exponentials
Let $X_1, X_2, X_3, ...$ be independent exponential random variables, each with parameter $λ = 1$. Compute the probability  
   $P(X_1 + X_2 + X_3 < 2\ \text{and}\ X_1 + X_2 + X_3 + X_4 > 2)$

## P9 Dice Duel: Expectation & Variance
Roll a fair die n times. Let $X$ be the number of 1's that are observed and $Y$ be the number of 2's that are observed. Answer the following questions:
(a) Show that: $Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)$  
(b) Find $ρ(X, Y)$.  
\(c) Find $E[XY]$.

## P10 Poisson Split Decision
Suppose that the number of customers visiting a fast food restaurant in a given day is $N ∼ Poisson(λ)$. Assume that each customer independently decides whether to purchase a drink, with probability $p$. Let $X$ be the number of customers who purchase a drink, and $Y$ be the number of customers who do not purchase a drink. That is, $X + Y = N$.
(a) Find the marginal distributions of $X$ and $Y$.  
(b) Find the joint PMF of $X$ and $Y$.  
\(c) Find the covariance $Cov(X, Y)$.  
(d) Find the correlation coefficient $ρ_{xy}$.  
(e) Are $X$ and $Y$ independent? Explain your reasoning.  
(f) Find $E[X^2Y^2]$.
