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Suppose we toss a coin 3 times. We denote head as "H" and tail as "T". We assume the probability of tail is P(T) = 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by P(X = k) for every possible value k. Check that k P(X= k) = 1. 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X.

Suppose we toss a coin 3 times. We denote head as "H" and tail as "T". We assume the probability of tail is P(T) = 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by P(X = k) for every possible value k. Check that k P(X= k) = 1. 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X.

A First Course in Probability (10th Edition)
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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Suppose we toss a coin 3 times. We denote head as “H” and tail as "T". We assume the probability of tail is P(T) = 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by P(X= k) for every possible value k. Check that ΣP(X = k) = 1. k 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X. 4. Do the followings with R: a. Create a plot of the pf of X. b. Create a plot of the cdf of X. c. Draw a random sample of size 1000 from Binomial(n = 3, p = 0.3). Then plot a histogram and calculate the sample mean and the sample variance. Comment briefly on these outputs compared to the results from question 1-3.
Transcribed Image Text:Suppose we toss a coin 3 times. We denote head as “H” and tail as "T". We assume the probability of tail is P(T) = 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by P(X= k) for every possible value k. Check that ΣP(X = k) = 1. k 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X. 4. Do the followings with R: a. Create a plot of the pf of X. b. Create a plot of the cdf of X. c. Draw a random sample of size 1000 from Binomial(n = 3, p = 0.3). Then plot a histogram and calculate the sample mean and the sample variance. Comment briefly on these outputs compared to the results from question 1-3.
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