Probability Theory Solutions
78 curated probability theory problems with marking points and complete derivations.
78 problems
Problem 1: Compute , , and ,
Let and be Poisson processes with parameters and , respectively, and suppose that and are mutually independent. Define . Compute , , and , . (Optional: Define .
Problem 2: Prove that follows a Poisson distribution with parameter
A counting process is defined to be a nonhomogeneous Poisson process with intensity function if: (ii) has independent increments Prove that follows a Poisson di
Problem 3: Prove that
Let be independent standard normal random variables. Prove that
Problem 4: Compute , where is a real number and is a non-negative integer;\\
The random variable follows an exponential distribution with parameter . Let denote the greatest integer not exceeding , and define the random variables .\\ (1)
Problem 5: Find
Consider a biased coin that lands heads with probability and tails with probability , where the outcomes are recorded as `` '' and `` '' respectively. The coin
Problem 6: Find the expectation and variance of
In a certain game, killing a particular monster has a fixed "drop rate" for equipment . Suppose that upon killing the monster, equipment drops with probability
Problem 7: Let the random variable follow an exponential distribution with parameter , and let follow an exponential…
Let the random variable follow an exponential distribution with parameter , and let follow an exponential distribution with parameter , where and are mutually i
Problem 8: Determine the value of
A random variable has probability density function , . (1) Determine the value of . (2) Find the distribution of . (3) Discuss whether is a continuous random ve
Problem 9: Find the distribution of ;
The random variables and have the joint density function . (1) Find the distribution of ; (2) Find the distribution of .
Problem 10: Determine the distribution of
Let the random variable follow the standard normal distribution . Given , the random variable follows the normal distribution . (1) Determine the distribution o
Problem 11: find the conditional probability of observing at least three tails;
A fair coin is tossed repeatedly until exactly two heads have appeared. Given that at least two tails were observed: (1) find the conditional probability of obs
Problem 12: determine the distribution of ;
Let the random variable follow the standard normal distribution , and let satisfy , with and independent. For the random variable : (1) determine the distributi
Problem 13: Find the density function of
(1) The random vector follows a multivariate normal distribution. For all , and . Find the density function of . (2) The random vector follows a normal distribu
Problem 14: Find the distributions of and
Let , and let . Find the distributions of and . Furthermore, determine whether and are independent.
Problem 15: Determine all possible distributions of
Let be a random variable. Suppose there exists a random variable , independent of , such that both and follow Poisson distributions. Determine all possible dist
Problem 16: Prove that follows a Poisson distribution with parameter
A counting process is said to be a non-homogeneous Poisson process with intensity function if: (ii) has independent increments Prove that follows a Poisson dist
Problem 17: prove that for , .\\
Let be independent and identically distributed random variables.\\ (1) If , prove that for , .\\ (2) If and , evaluate .
Problem 18: The random variable follows an exponential distribution with parameter , and follows an exponential distribution with…
The random variable follows an exponential distribution with parameter , and follows an exponential distribution with parameter , where and are independent and
Problem 19: Prove that: (1)
Let be i.i.d. random variables satisfying , and let . Prove that: (1) (2)
Problem 20: Determine whether the simple random walk on is recurrent or transient, and prove your conclusion
Consider an infinite connected graph with uniformly bounded vertex degrees. Define , where if and only if and there exists such that Determine whether the simpl
Problem 21: find the probability that both children are girls
A family has two children whose genders and birth order are unknown. (1) Given that the older child is a girl, find the probability that both children are girls
Problem 22: Prove that the random variables converge in distribution to
Suppose the random variables converge in distribution to , and converge in distribution to a positive constant . Prove that the random variables converge in dis
Problem 23: Find and
Consider a biased coin that lands heads with probability and tails with probability , where the outcomes are recorded as `` '' for heads and `` '' for tails. Th
Problem 24: Let
Let . Using a probabilistic method, evaluate: .
Problem 25: find the probability that all ducks fall within the same semicircle
This problem (7+8 points) aims to discuss the probability that ducks swimming independently in a circular pond all fall within the same semicircle. Let their po
Problem 26: Prove that
Suppose is a function of period 1 that is continuous on . Let be a random variable uniformly distributed on , and let be an irrational number. Define . Prove th
Problem 27: Find the distributions of and
Let , and define , . Find the distributions of and . Furthermore, determine whether and are independent.
Problem 28: Prove that a random variable is independent of itself if and only if is almost surely…
Prove that a random variable is independent of itself if and only if is almost surely constant.
Problem 29: Prove that defines a distance (metric) on the space of random variables
For random variables , define (1) Prove that defines a distance (metric) on the space of random variables. That is, show that satisfies positive definiteness, s
Problem 30: Prove that converges in probability to 1
Let follow the uniform distribution on , and define Prove that converges in probability to 1. (20 points)
Problem 31: Compute , , and , (Optional: Define
Let and be Poisson processes with parameters and , respectively, and suppose that and are mutually independent. Define . Compute , , and , (Optional: Define . C
Problem 32: Determine the distribution of
Let the random variable follow the standard normal distribution , and let satisfy , with and mutually independent. For the random variable : (1) Determine the d
Problem 33: Find the probability that these particles meet infinitely often
Let . Suppose particles independently start simultaneously from the origin and perform one-dimensional simple random walks. We say the particles meet whenever a
Problem 34: prove that\\
Let the random variables be independent and identically distributed with the exponential distribution of parameter 1. Given a positive real number , prove that\
Problem 35: Find the distributions of and
Let , and let . Find the distributions of and . Furthermore, determine whether and are independent.
Problem 36: Determine the distribution of ;
The random variable follows the standard normal distribution . Given , the random variable follows the normal distribution . (1) Determine the distribution of ;
Problem 37: Prove that a random variable is independent of itself if and only if is almost surely…
Prove that a random variable is independent of itself if and only if is almost surely constant.
Problem 38: determine the probability that all ducklings fall within the same semicircle
7+8 pts) This problem discusses the probability that ducklings swimming independently in a circular pond all fall within the same semicircle. Suppose their posi
Problem 39: prove that
Let the random variables be independent and identically distributed, each following an exponential distribution with parameter 1. Given a positive real number ,
Problem 40: prove that for , .\\
Let be independent and identically distributed random variables.\\ (1) If , prove that for , .\\ (2) If and , compute .
Problem 41: Compute , where is a real number and is a nonnegative integer;\\
The random variable follows an exponential distribution with parameter . Let denote the greatest integer not exceeding , and define the random variables .\\ (1)
Problem 42: find the probability that both children are girls
A family has two children whose genders and birth order are unknown. (1) Given that the older child is a girl, find the probability that both children are girls
Problem 43: Prove that for
Consider the simple symmetric random walk , where each step increment satisfies . The initial position is . Let . Prove that for .
Problem 44: Prove that
Let be i.i.d. with . Define . Prove that .
Problem 45: Prove that defines a metric on the space of random variables
For random variables , define (1) Prove that defines a metric on the space of random variables. That is, prove that satisfies positive definiteness, symmetry, a
Problem 46: Determine all possible distributions of
Let be a random variable. There exists a random variable , independent of , such that both and follow Poisson distributions. Determine all possible distribution
Problem 47: prove that
The random variables are independent and identically distributed, each following an exponential distribution with parameter 1. Given a positive real number , pr
Problem 48: prove that for , .\\
Let be independent and identically distributed random variables.\\ (1) If , prove that for , .\\ (2) If and , compute .
Problem 49: Compute the density function of
(1) The random vector follows a normal distribution. For all , . Compute the density function of . (2) The random vector follows a normal distribution with mean
Problem 50: find:
Let . Using a probabilistic method, find: .
Problem 51: Prove that the random variable converges in distribution to
The random variable converges in distribution to , and converges in distribution to a positive constant . Prove that the random variable converges in distributi
Problem 52: Find the distribution of ;
The random variables and have joint density function . (1) Find the distribution of ; (2) Find the distribution of .
Problem 53: Prove that
Let follow the standard normal distribution. Prove that
Problem 54: Find
Consider a biased coin that lands heads with probability and tails with probability . We use `` '' and `` '' to record heads and tails, respectively. The coin i
Problem 55: Prove that there exist constants and such that converges in distribution to the standard normal distribution
Let be independent and identically distributed with . Define Let . Prove that there exist constants and such that converges in distribution to the standard norm
Problem 56: Find the expectation and variance of
In a certain game, killing a particular monster has a fixed "drop rate" for producing items . Suppose that after killing the monster, item drops with probabilit
Problem 57: Prove that defines a metric on random variables, i.e., satisfies positive definiteness, symmetry, and the triangle…
For random variables , define (1) Prove that defines a metric on random variables, i.e., satisfies positive definiteness, symmetry, and the triangle inequality;
Problem 58: Prove that a random variable is independent of itself if and only if is almost surely…
Prove that a random variable is independent of itself if and only if is almost surely constant.
Problem 59: Compute the distribution of ;
The random variable follows the standard normal distribution , satisfies , and are independent. For the random variable : (1) Compute the distribution of ; (2)
Problem 60: Determine the distribution of ;
The random variable follows the standard normal distribution . Given , the random variable follows the normal distribution . (1) Determine the distribution of ;
Problem 61: Prove that is also a stopping time with respect to
Let be a filtration (an increasing sequence of -fields). Let be a stopping time with respect to , and let be an integer-valued random variable that is -measurab
Problem 62: find the probability that all ducklings fall within the same semicircle
(7+8 points) This problem discusses the probability that ducklings swimming independently in a circular pond all fall within the same semicircle. Suppose their
Problem 63: Compute the conditional probability of tossing at least three tails;
A fair coin is tossed repeatedly until two heads are obtained. Given that at least two tails were tossed: (1) Compute the conditional probability of tossing at
Problem 64: Compute the value of ;
The random variable has probability density function , . (1) Compute the value of ; (2) Determine the distribution of ; (3) Discuss whether is a continuous rand
Problem 65: Compute , where is a real number and is a nonnegative integer;\\
A random variable follows an exponential distribution with parameter . Let denote the greatest integer not exceeding , and define the random variables .\\ (1) C
Problem 66: Determine all possible distributions of
Let be a random variable. There exists a random variable independent of such that both and follow Poisson distributions. Determine all possible distributions of
Problem 67: Compute the conditional probability of obtaining at least three tails;
A fair coin is tossed repeatedly until two heads have appeared. Given that at least two tails have occurred: (1) Compute the conditional probability of obtainin
Problem 68: Prove that converges in probability to 1
Let be uniformly distributed on , and define Prove that converges in probability to 1. (20 points)
Problem 69: Compute the value of ;
A random variable has probability density function , . (1) Compute the value of ; (2) Determine the distribution of ; (3) Discuss whether is a continuous random
Problem 70: Let
Let . Using probabilistic methods, evaluate: .
Problem 71: Let be independent of the -field filtration , with
Let be independent of the -field filtration , with . Let be a stopping time with respect to the filtration , and . Then .
Problem 72: Prove that
Given i.i.d. with . Let . Prove that .
Problem 73: Find the distribution of the total time spent at the root vertex
On the 3-regular tree, consider the jump process starting from the root, where the transition rate along every edge is 1. Find the distribution of the total tim
Problem 74: Find the distribution of ;
The random variables and have the joint density function . (1) Find the distribution of ; (2) Find the distribution of .
Problem 75: Prove: (1)
Let be i.i.d. random variables satisfying , and denote . Prove: (1) (2)
Problem 76: The random variable follows an exponential distribution with parameter , and follows an exponential distribution with…
The random variable follows an exponential distribution with parameter , and follows an exponential distribution with parameter . The two are mutually independe
Problem 77: Prove that the random variables converge in distribution to
The random variables converge in distribution to , and converge in distribution to a positive constant . Prove that the random variables converge in distributio
Problem 78: find the probability that both children are girls
A family has two children whose genders and birth order are unknown. (1) Given that the older child is a girl, find the probability that both children are girls