Understanding Discrete Random Variables and Their Probabilities
School
University of Ontario Institute of Technology**We aren't endorsed by this school
Course
ELEE 2110
Subject
Statistics
Date
Dec 11, 2024
Pages
10
Uploaded by ConstableAlbatross1680
17.1) In a bag, there are 12 identical balls, numbered 1 to 12 inclusive. Let 𝑋be the discrete random variable denoting the ball drawn from the bag. Determine the conditional pmf of 𝑋given 𝐵, where 𝐵is the event representing balls with prime numbers. Solution We have 𝐵 ൌ ሼ2, 3, 5, 7, 11ሽ → 𝑃ሺ𝐵ሻ ൌ512.The conditional pmf would be as follows: 𝑝ሺ2|𝐵ሻ ൌ 𝑝ሺ3|𝐵ሻ ൌ 𝑝ሺ5|𝐵ሻ ൌ 𝑝ሺ7|𝐵ሻ ൌ 𝑝ሺ11|𝐵ሻ ൌ112512ൌ15.
17.2) The number of text messages sent by a teenager during an hour is a random variable. The mean and variance of this random variable are 15and 9, respectively. Using the Chebyshev inequality, estimate the probability that the number of text messages is more than 5from the mean. Solution Using the Chebyshev inequality with 𝐸ሾ𝑋ሿ ൌ 15, 𝜎 ൌ 3, and 𝑐 ൌ 5, we have the following: 𝑃ሾ|𝑋 െ 15| 5ሿ 925ൌ 0.36.
17.3) Assume 1000 bits are independently transmitted over a digital communications channel in which the bit error rate is 0.001. Determine the probability when the total number of errors is greater than or equal to 998. Solution Using Binomial distribution, we have 𝑃ሺ998 𝑋 1000ሻ ൌ ൬1000𝑥൰ ሺ0.001ሻ௫ሺ1 െ 0.001ሻଵି௫ଵ௫ୀଽଽ଼.
17.4) Suppose phone call arrivals at a call center are Poisson and occur at an average rate of 50 per hour. The call center has only one operator. If all calls are assumed to last one minute, determine the probability that a waiting line will occur. Solution A waiting line will occur if two or more calls arrive in any one-minute interval. This event is a complement to the event that no call or one call arrives. Noting the arrival rate 𝜆is ହcalls per minute and 𝑥 1, we have the following: Probability of a waiting line ൌ 1 െ𝑒ିହቀ56ቁ0!െ𝑒ିହቀ56ቁଵ1!≅ 0.20324.
17.5) The variance of the discrete uniform random variable 𝑍, which takes on values in a set of 𝑛consecutive integers, is 4. Determine the mean of this random variable. Solution The variance is as follows: 𝑛ଶെ 112ൌ 4 → 𝑛 ൌ 7we thus get the following mean ሺ𝑗 1ሻ ሺ𝑗 7ሻ2ൌ 𝑗 4where 𝑗is an integer.
17.6) The number of major earthquakes in the world is represented by a Poisson distribution with a rate of 𝜆 ൌ 7.4earthquakes in a year. Determine the probability that there are exactly four earthquakes in a year. What is the probability that there are no earthquakes given that there are at most two earthquakes in a year? Solution We have 𝜆 ൌ 7.4and 𝑥 ൌ 4, we thus have 𝑃ሺ𝑋 ൌ 4ሻ ൌ𝑒ି.ସ7.4ସ4!≅ 0.075.We have 𝑃ሺ𝑋 ൌ 0|𝑋 2ሻ ൌ𝑃ሺ𝑋 ൌ 0, 𝑋 2ሻ𝑃ሺ𝑋 2ሻൌ𝑃ሺ𝑋 ൌ 0ሻ𝑃ሺ𝑋 2ሻൌ𝑒ି.ସ7.40!𝑒ି.ସ7.40!𝑒ି.ସ7.4ଵ1!𝑒ି.ସ7.4ଶ2!≅ 0.028
17.7) Consider a six-sided cube-shape die, which is not fair. Let 𝑋be the discrete random variable which represents the outcome of a roll of the die. The pmf of the random variable 𝑋is as follows: 𝑃ሺ𝑋 ൌ 1ሻ ൌ 𝑃ሺ𝑋 ൌ 2ሻ ൌ 𝑃ሺ𝑋 ൌ 3ሻ ൌ 𝑃ሺ𝑋 ൌ 5ሻ ൌ 𝑃ሺ𝑋 ൌ 6ሻ ൌ 0.1 & 𝑃ሺ𝑋 ൌ 4ሻ ൌ 0.5.Determine the variance of the random variable 𝑋. Solution 𝐸ሾ𝑋ሿ ൌ ሺ1ሻሺ0.1ሻ ሺ2ሻሺ0.1ሻ ሺ3ሻሺ0.1ሻ ሺ4ሻሺ0.5ሻ ሺ5ሻሺ0.1ሻ ሺ6ሻሺ0.1ሻ ൌ 3.7𝐸ሾ𝑋ଶሿ ൌ ሺ1ଶሻሺ0.1ሻ ሺ2ଶሻሺ0.1ሻ ሺ3ଶሻሺ0.1ሻ ሺ4ଶሻሺ0.5ሻ ሺ5ଶሻሺ0.1ሻ ሺ6ଶሻሺ0.1ሻ ൌ 15.5𝜎ଶൌ 𝐸ሾ𝑋ଶሿ െ ሺ𝐸ሾ𝑋ሿሻଶൌ 15.5 െ 3.7ଶൌ 1.81
17.8) Suppose the average age of people in a town is 49 years. If all people over 70 years old should be vaccinated against a certain disease, determine the maximum fraction of people of the town who should be vaccinated using the Markov inequality. Solution 𝑃ሺ𝑋 𝑐ሻ 𝐸ሾ𝑋ሿ𝑐→ 𝑃ሺ𝑋 70ሻ 4970ൌ 70%.
17.9) In a typical lottery game, a player chooses 𝑛distinct numbers from 1 to 𝑁inclusive, where 𝑛and 𝑁are both positive integers. Determine the probability that 0 𝑘 𝑛of the 𝑛balls picked match the player’s choices. Assuming 𝑁 ൌ 49and 𝑛 ൌ 6, determine the specific probabilities of winning for various values of 𝑘. Solution Using hypergeometric probability, we have the following probability: 𝑝 ൌ൫൯൫ேିି൯൫ே൯.We thus have the following probabilities for 𝑁 ൌ 49, 𝑛 ൌ 6, and various values of 𝑘: 𝑘𝑝0 43.5965%1 41.3019%2 13.2378%3 1.7650%4 0.0969%5 0.0018%6 0.000007%
17.10) Suppose 𝑋is a discrete random variable represented by the outcome of rolling a fair typical die (i.e., its six outcomes 1, 2, 3, 4, 5, and 6 are equally likely) and the random variable 𝑌is related to 𝑋by the deterministic function 𝑌 ൌ cos ቀగଷቁ. Determine the sample space, pmf, and mean value of the random variable 𝑌. Solution The sample space for the discrete random variable 𝑋is 𝑆ൌ ሼ1, 2, 3, 4, 5, 6ሽ, we thus have the following: 𝑋 ൌ 1 → 𝑌 ൌ cos ቀ𝜋3ቁ ൌ12.𝑋 ൌ 2 → 𝑌 ൌ cos ൬2𝜋3൰ ൌ െ12.𝑋 ൌ 3 → 𝑌 ൌ cos ൬3𝜋3൰ ൌ െ1.𝑋 ൌ 4 → 𝑌 ൌ cos ൬4𝜋3൰ ൌ െ12.𝑋 ൌ 5 → 𝑌 ൌ cos ൬5𝜋3൰ ൌ12.𝑋 ൌ 6 → 𝑌 ൌ cos ൬6𝜋3൰ ൌ 1.Consequently, the sample space for the random variable 𝑌is as follows: 𝑆ൌ ൜െ1, െ12,12, 1ൠ.Noting that 𝑝൫𝑥൯ ൌଵ, 𝑗 ൌ 1, 2, … , 6, and 𝑌is a one-to-one function, 𝑝ሺ𝑦ሻis as follows: 𝑝ሺെ1ሻ ൌ16, 𝑝ሺ1ሻ ൌ16, 𝑝൬െ12൰ ൌ 𝑝൬12൰ ൌ13.We thus have 𝐸ሾ𝑌ሿ ൌ െ1 ൈ16െ12ൈ1312ൈ13 1 ൈ16ൌ 0.