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中文题目A message must traverse a chain of relay nodes to reach its destination. Each node independently takes $\operatorname{Geom}(1/3)$ attempts to successfully forward the message to the next node. However, on each attempt, there is an independent probability $1/5$ that the node perma
打开 →Let $X$ have the mixture density $f(x) = \frac{1}{2}e^{-x} + \frac{5}{2}e^{-5x}$ for $x \geq 0$ (a $50{-}50$ mixture of $\operatorname{Exp}(1)$ and $\operatorname{Exp}(5)$). (a) Compute $P(X > s + t \mid X > s)$ as a function of $s$ and $t$, and show it depends on $s$ (i.e., the
打开 →Let $X \sim \text{Geometric}(p)$ count the number of independent Bernoulli($p$) trials until the first success (so $P(X = k) = (1-p)^{k-1} p$ for $k = 1, 2, \ldots$). (a) Derive a closed-form expression for $P(X > n)$. (b) Prove the memorylessness property: for all positive int
打开 →Let $X \sim \text{Exponential}(\lambda)$. Prove that $P(X > s + t \mid X > s) = P(X > t)$ for all $s, t \geq 0$. Then show the exponential is the only continuous distribution with this property.
打开 →Let $X \sim \operatorname{Exp}(2)$. Using the memoryless property, compute $E[X \mid X > 3]$.
打开 →A device's lifetime $X$ has survival function $\bar{F}(t) = P(X > t)$ and hazard rate $h(t) = f(t)/\bar{F}(t)$. Show that the memoryless property $P(X > s + t \mid X > s) = P(X > t)$ implies $h(t) = \lambda$ (a constant) for all $t \geq 0$, and conversely that a constant hazard r
打开 →A factory runs 3 identical machines with independent lifetimes $\operatorname{Exp}(1)$. When any machine fails, it is instantly replaced with a new identical machine. All non-failed machines continue running (their residual lifetimes remain $\operatorname{Exp}(1)$ by memorylessne
打开 →A system has 4 independent components, each with lifetime $\operatorname{Exp}(2)$. When a component fails, it is removed and the remaining components continue operating. By memorylessness, surviving components' residual lifetimes are still $\operatorname{Exp}(2)$. Find the expect
打开 →Part (a): Let $X$ be a continuous, positive random variable satisfying $P(X > s + t \mid X > s) = P(X > t)$ for all $s, t \geq 0$. Prove that $X$ must be exponentially distributed. Part (b): A lightbulb's lifetime $L$ has CDF $F(t) = 1 - \frac{1}{2}e^{-t} - \frac{1}{2}e^{-3t}$ f
打开 →A radioactive atom has a lifetime $X \sim \operatorname{Exp}(1/2)$. Given that the atom has survived past time $t = 3$, what is the probability it survives past time $t = 7$?
打开 →A bus arrives at a stop at an $\operatorname{Exp}(1/10)$ random time (mean 10 minutes). You have already been waiting for 5 minutes. What is the expected additional waiting time?
打开 →Part (a): Let $N$ be a positive-integer-valued random variable satisfying $P(N > m + n \mid N > m) = P(N > n)$ for all $m, n \in \mathbb{Z}_{\geq 0}$. Prove that $N$ must follow a geometric distribution. Part (b): For $N \sim \operatorname{Geom}(p)$, compute $E[N^2 \mid N > k]$
打开 →A full-attention model uses L=1024 tokens and stores one attention score matrix per head in float16. Roughly how much memory does one head's score matrix use?
打开 →Two independent alarms go off at $\operatorname{Exp}(4)$ and $\operatorname{Exp}(6)$ times respectively. If alarm 1 fires first you pay $\$3$; if alarm 2 fires first you pay $\$5$. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an addit
打开 →Let $X \sim \operatorname{Exp}(\lambda)$. Using the memoryless property, find $\operatorname{Var}(X \mid X > t)$ for $t > 0$. Does conditioning on survival change the variance compared to $\operatorname{Var}(X)$? Evaluate numerically for $\lambda = 5$ and $t = 2$.
打开 →Four independent exponential random variables $X_1 \sim \operatorname{Exp}(1)$, $X_2 \sim \operatorname{Exp}(2)$, $X_3 \sim \operatorname{Exp}(3)$, $X_4 \sim \operatorname{Exp}(6)$ represent task completion times. Using iterated applications of the memoryless property, find $P(X_
打开 →Let $X \sim \operatorname{Geom}(1/4)$ (number of trials until first success). Using the memoryless property of the geometric distribution, compute (i) $E[X \mid X > 5]$ and (ii) $P(X > 8 \mid X > 5)$.
打开 →Let $X_1, X_2, X_3$ be independent, each $\operatorname{Exp}(4)$. Find the distribution of $M = \min(X_1, X_2, X_3)$ and compute $E[M]$. Then verify: given that $M > 2$, use memorylessness to find $E[M \mid M > 2]$.
打开 →Three players with independent lifetimes $X_1 \sim \operatorname{Exp}(1)$, $X_2 \sim \operatorname{Exp}(2)$, $X_3 \sim \operatorname{Exp}(4)$ compete. The first to "die" is eliminated, then the two survivors continue (by memorylessness, their residual lifetimes are fresh exponent
打开 →A gambler plays a sequence of rounds. In each round, he flips a biased coin with $P(\text{heads}) = p$ repeatedly until he gets heads; the number of flips in that round is $\operatorname{Geom}(p)$. The number of rounds he plays is itself $\operatorname{Geom}(q)$ (independent of t
打开 →A machine has two critical components in series: component A with lifetime $\operatorname{Exp}(3)$ and component B with lifetime $\operatorname{Exp}(5)$, independent of each other. When either fails, the entire machine stops, the failed component is replaced (cost $\$20$ for A, $
打开 →You flip a coin with $P(\text{heads}) = 1/3$ until you get heads. Given that the first 8 flips were all tails, what is the expected total number of additional flips needed (starting from flip 9)?
打开 →You roll a fair die repeatedly until you roll a 6. Let $N$ be the number of rolls needed. Given that you have not yet rolled a 6 in the first 5 rolls, what is the probability that you will need more than 10 rolls total?
打开 →Let $X \sim \operatorname{Exp}(\lambda)$ and $Y \sim \operatorname{Exp}(\mu)$ be independent. Player A finishes at time $X$ and player B finishes at time $Y + c$ where $c > 0$ is a head start for player A (player B starts $c$ time units later). (a) Derive $P(X < Y + c)$ — the pr
打开 →Why can two jump processes share exactly the same jump chain but still look very different when observed in real time?
打开 →Why can a state have a small stationary probability even if it has a large exit rate?
打开 →Let $X \sim \operatorname{Exp}(2)$ and $Y \sim \operatorname{Exp}(3)$ be independent. Define $M = \min(X, Y)$ and fix a threshold $c = 1$. (i) Find $P(M > 1)$. (ii) Given $M > 1$, find $E[M - 1 \mid M > 1]$ and $P(X < Y \mid M > 1)$ (the probability that $X$ is the minimum, giv
打开 →Why is first-step analysis so effective for expected hitting times in jump processes?
打开 →A simulator gets the jump-chain routing probabilities right but replaces exponential waits by deterministic one-minute waits in every state. Why is the resulting calendar-time process generally not a CTMC anymore?
打开 →Why can a CTMC with different exit rates across states still be simulated using one common Poisson clock plus occasional virtual self-jumps?
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