2)
expectation value of X =<X>=xP(x)=1*(1/6)+2*(1/6)+3*(1/6)+4*(1/6)+5*(1/6)+6*(1/6)=21/6
<X2> =x2P(x)=12*(1/6)+22*(1/6)+32*(1/6)+42*(1/6)+52*(1/6)+62*(1/6)=91/6
expectation of X- =<X->
=(1-21/6)*(1/6)+(2-21/6)*(1/6)+(3-21/6)*(1/6)+(4-21/6)*(1/6)+(5-21/6)*(1/6)+(6-21/6)*(1/6)=0
Example: Consider our coin example again, with X-1, each outcome with probability P(X- 1/2. The expectation...
i. Consider a weighted 6-sided die that is twice as likely to produce any even outcome as any odd outcome. What is the expected value of 1 roll of this die? What is the expected value of the sum of 9 rolls of this die? ii. Let X denote the value of the sum of 10 rolls of an unweighted 6-sided die. What is Pr(X = 0 mod 6)? (Hint: it is sufficient to consider just the last roll) *side...
3. (25 pts) A Truly FAIR COIN: Because actual coins are not truly balanced, P - the ACTUAL probability of HEAD for our old, battered coin - may differ substantially from 1/2. The famous Mathematician John Von-Neumann came up with the following proposal for using our possibly unfair coin to simulate a truly fair coin that always has PROB(HEAD)=PROB(TAIL) = 1/2, as follows: • (i) toss the UNFAIR coin twice. This is the experiment E. • (ii) IF you got...
Problem 6 7. Part 1-4 (10pts)A coin has two faces Head and Tail. (1) (2pts)]lf you toss the coin once, and record the up-face value, what is the sample space? 6. (2) (2pts)lf you toss the coin once, what is the probability that up-face is Tail? What is the probability that up-face is Head? (3) (5ps)lf you toss the coin three times, and record the up-face value for each toss. One of the possible outcome is (Head, Head, Head). By...
Probability calculation: discrete/finite example • Two rolls of a tetrahedral die . Let every possible outcome have probability 1/16 • P(X = 1) = Let 2 = min(X,Y) Y = Second roll • P(Z = 4) = • P(Z = 2) = 1 4 2 3 X = First roll
We have four fair coins, each of which has probability 1/2 of having a heads outcome and a tails outcome. The experiment is to ip all four coins and observe the sequence of heads and tails. For example, outcome HTHH means coin 1 was heads, coin 2 was tails, coin 3 was heads, coin 4 was heads Note that there are 16 total outcomes, and we assume that each one is equally likely. What is the probability that at there...
Exercise 1. C What is the expectation of X in Example 2 if the number of days until the virus is stopped is Geometric with expectation of 4 days? Example 2. C A computer virus spreads at a rate of 25% per day, and the number of days until it is stopped is Geometric with expectation of 5 days. What is the expected number of computers affected until it is stopped, if on day zero 4000 computers were affected? Let...
3 Probability and Statistics [10 pts] Consider a sample of data S obtained by flipping a coin five times. X,,i e..,5) is a random variable that takes a value 0 when the outcome of coin flip i turned up heads, and 1 when it turned up tails. Assume that the outcome of each of the flips does not depend on the outcomes of any of the other flips. The sample obtained S - (Xi, X2,X3, X, Xs) (1, 1,0,1,0 (a)...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. Assume that n-2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes xi and x2 Let estimate μ X of true mean #xbe μχ = (x1+x2)/2. Then the random variable associated with estimate μ xis estimator random 1. a. Show the...
Hello, There is an example in my probability textbook that asks us to calculate the probability of throwing exactly two heads in three tosses using a balanced coin. The reasoning the textbook uses is written below, but our professor wants us to criticize their reasoning, stating that it is a common mistake seen in introductory probability textbooks. They argue that just because the coin is balanced, each outcome (the result of three tosses) must have probability 1/8. Note that a...
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