4. (Random Variables). (i) A coin weighted so that P(H) = 14/31 and P(T) = 17/31 is tossed four times. Let X be the randomvariable which denotes the length of the longest string of heads which occurs. Find the distribution, expectation, variance and standarddeviation of X.(ii) A fair coin is tossed until a head or 6 tails occur. Let X be the random variable which denotes the number of tosses. Find thedistribution, expectation, variance and standard deviation of X.(iii) A player tosses two fair dice. If the sum is prime, he wins that number of dollars, but otherwise he loses that number of dollars.Find the expected value of the game, and determine which of the following is true: (a) the game is favorable for the player, (b) thegame is unfavorable for the player, (c) the game is fair.

Answered: Image /qna-images/answer/28d0a24e-90c1-4bce-816c-e7021863b38a.jpg

Answered: 4. (Random Variables). (i) A coin…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Assume that the amount of weight that male college students gain during their freshman year are normally distributed with a. Mean of u= 1.2 kg and a standard deviation of o= 5.1 kg A. If 1 male college student is randomly selected, Find the probability that he gains between 0 and 3 kg during freshman year B. If 25 male college students are randomly selected find the probability that their mean weight gain during freshman year is between 0 and 3 kg why can the normal distribution be used on pet b even though the sample size does not exceed 30?
В2. (a) Given the probability Venn diagram below for the two Events A and B find the probabilities: 0.6 0.5 A 0.2 (i) (ii) (iii) Pr( A 'OR' B) Pr( Not-A 'AND' Not-B) Pr( Not-A "OR' Not-B) (b) A random sample from a population of X with unknown mean but a known variance, o = 400, gives the following data: sample size, n= 125, Ex= 1750. (i) Give a point estimate of the population mean, u. (ii) Is this estimate biased or not. Give a reason. (ii) Derive a 95% confidence interval for the population mean, µ.
A population has a mean of 200 and a standard deviation of 50. Suppose a sample of size 100 is selected and x is used to estimate u. (Round your answers to four decimal places.) (a) What is the probability that the sample mean will be within +5 of the population mean? (b) What is the probability that the sample mean will be within +10 of the population mean?
Use this information about the overhead reach distances of adult females: µ = 205.5 cm, σ = 8.6 cm, and overhead reach distances are normally distributed (based on data from th Federal Aviation Administration) If 40 adult females are randomly selected, find the probability that they have a mean overhead reach between 204.0 cm and 206.0 cm.
You are to roll a fair die n = 120 times, each time observing if the topside of the die shows a 6 (success) or not (failure). After observing the n = 120 tosses, you are to count the number of times the topside showed a 6. This count is represented by the random variable X. (a) The distribution of X ? with a mean and a standard deviation Enter your answer using all the decimals you can. (b) Now think about the proportion of your n = 120 tosses that show a six. What can you say about the distribution of this proportion? Complete the sentence, Enter your answer using all the decimals you can. The distribution of ? with a mean and a standard deviation σ you can. ☐ 2 2 ☐ (c) What is the probability that the proportion/percentage of your n = 120 tosses that show a six will be somewhere between 15% and 22%? Enter your answer using all the decimals you can. (d) After the n = 120 tosses of the die, you observe X = 27, the value of the sample proportion is then = 270 = 0.225. What is the…
If a discrete random variable X has the following probability distribution: X -2, - 1, 0, 1, 2 P(X) 0.2, 0.3, 0.15, 0.2, 0.15 Use this to find the following: (a) The mean of X and E[X^2]. (b) The probability distribution for Y = 2X^2 + 2 (i.e, all values of Y and P(Y )). (c) Using part (b) (i.e, the probability distribution forY ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. **Note: Please do not just copy from Chegg!
Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of seven days. (a) In words, define the random variable X. Edit Insert - Formats - B A く) (b) Find the ZSCore for a trial lasting 20 days. (Round your answer to three decimal places.) (c) If one of the trials is randomly chosen, find the probability that it lasted at least 20 days. (d) Shade the area corresponding to this probability in the graph below. (Hint: The x-axis is the z-score. Use your z-score from part (b), rounded to one decimal place).
I have a probability distribution consisting of (x, p(x)): (-3, 0.1), (-2, 0.2), (-1, 0.3), (0, 0.05). The probabilities obviously do not sum to 1. How do I find <x> (average), <x2> (the second moment), <sigmax2> (the variance)?
Fast Transport determined that on an annual basis, the distance travel per lorry is normally distributed, with a mean of 50 thousand kilometers and a standard deviation of 10 thousand kilometers. Compute the probability that, (i). a randomly selected lorry travels from 36 to 50 thousand kilometers in a year. (ii). a randomly selected lorry travels more than 45 thousand kilometers in a year. (iii). the mean of the sample will be less than 48 thousand kilometers if a sample of 50 lorries is selected.
X is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 20 items from X and you calculate the average of this sample. What is the expected value of this sample average? (i.e. what value would you expect to get for the average of this small sample?
Q7) The length of time, in seconds, that a computer user takes to read his or her e- mail is distributed as lognormal random variable with μ = 1.8 and o² = 4. (a) (b) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute? What is the probability that a user reads e-mail for a length of time that is equal to the mean of the underlying lognormal distribution?
3. Table 1 below shows the probability distribution of hurricanes that have made landfall in the Caribbean. The random variable, X, is the category of the hurricane. Category 1 is the weakest hurricane level while category 5 is the strongest. Table 1 x 1 2 3 4 5 P(X=x) 0.41 0.34 0.13 0.07 * a) Find the value of ‘*’. [1] b) Find the expected category of hurricane to make landfall, E(X). [2] c) What is the probability that a Category 4 or stronger hurricane will make landfall?[1] d) Calculate the variance. [3]

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