A fair die is thrown. If the number on the die is 2 or less, a...
A fair die is thrown. If the number on the die is 2 or less, a random variable X is assigned the value 1 and the value zero otherwise. The random variable X is then input to a binary channel with crossover probabilities v = 0.001 and e = 0.003. The output of the binary channel is the random variable Y. Find the following:
(a) Pr[ '3' | X = 0] or the probability that the die throw is 3 given that the random variable X is zero.
(b) Pr[X = 0|Y = 0] or the probability that X = 0 given that we observe Y = 0.
Homework Answers
A fair coin is tossed. If the toss results in a head, then one die is...
A fair coin is tossed. If the toss results in a head, then one die is thrown, while if the toss results in a tail, then two dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The values that X can assume are the positive integers from 1 to 12 inclusive. Find the following probabilities. Your answers should be whole numbers or fractions in lowest terms. Pr(X=1)...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] =...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
A fair die is rolled 3 times. If 1 or 6 lands uppermost in a trail,...
A fair die is rolled 3 times. If 1 or 6 lands uppermost in a trail, then the throw is considered a success. Otherwise, the throw is considered a failure. What is the probability of obtaining 0 or 1 success in the experiment? Round your answer to two decimal places. Question 3 1 pts A fair die is rolled 3 times. If 1 or 6 lands uppermost in a trail, then the throw is considered a success. Otherwise, the throw...
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair...
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
7. The input U of binary communication channel is either -2.5 or +2.5 representing bit values...
7. The input U of binary communication channel is either -2.5 or +2.5 representing bit values b = 0 and b = 1 respectively, where P(b = 1) = 0.75. The channel output is given by V = U+N where “channel noise” N is a continuous random variable whose pdf is a symmetric triangular function in the range (-3, +3). Assume that U and N are independent. The receiver decodes the channel output to produces a bit value b as...
3. Compute the variance of X when X is the result of rolling a fair die....
3. Compute the variance of X when X is the result of rolling a fair die. 4. Let X be a random variable with density function 1 0<z<1 0 otherwise f(x)= What is the variance of X.
randi (Lo), 1000, Lab Description In this lab, you are to simulate a simple binary communication...
randi (Lo), 1000, Lab Description In this lab, you are to simulate a simple binary communication channel characterized by appropriate conditional and prior probabilities and estimate the probability of error as well as the probability of receiving either a 1 or a 0. Start with a symmetrie binary communication channel characterized by the conditional probabiliti PLRlis 0.995 and POR I PR 0.003 The prior prebabilities of a 0 or a l are given by POs)-0.389 a PI ls 0.6 Xi...
help with number 4 4. Roll a die and flip a coin. Let Y be the...
help with number 4
4. Roll a die and flip a coin. Let Y be the value of the die. Let Z = 1 if the coin shows a head, and Z = 0 otherwise. Let X = Y + Z. Find the variance of X. 5. (a) If X is a Poisson random variable with = 3, find E(5*).
Use a random number generator to simulate the roll of a fair die 100 times. Let...
Use a random number generator to simulate the roll of a fair die 100 times. Let the number face up on the die represent the variable X A. Build a relative frequency table of the outcomes of the variable X. X Freq Rel. Freq B. Use the relative frequency distribution from part c to estimate the probability of an even number face up, then find the actual probability using the probability distribution and comment on the difference in values.
A fair coin is tossed 20 times. Let X be the number of heads thrown in...
A fair coin is tossed 20 times. Let X be the number of heads thrown in the first 10 tosses, and let Y be the number of heads tossed in the last 10 tosses. Find the conditional probability that X = 6, given that X + Y = 10.
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
A fair die is rolled 3 times. If 1 or 6 lands uppermost in a trail, then the throw is considered a success. Otherwise, the throw is considered a failure. What is the probability of obtaining 0 or 1 success in the experiment? Round your answer to two decimal places. Question 3 1 pts A fair die is rolled 3 times. If 1 or 6 lands uppermost in a trail, then the throw is considered a success. Otherwise, the throw...
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
7. The input U of binary communication channel is either -2.5 or +2.5 representing bit values b = 0 and b = 1 respectively, where P(b = 1) = 0.75. The channel output is given by V = U+N where “channel noise” N is a continuous random variable whose pdf is a symmetric triangular function in the range (-3, +3). Assume that U and N are independent. The receiver decodes the channel output to produces a bit value b as...
3. Compute the variance of X when X is the result of rolling a fair die. 4. Let X be a random variable with density function 1 0<z<1 0 otherwise f(x)= What is the variance of X.
randi (Lo), 1000, Lab Description In this lab, you are to simulate a simple binary communication channel characterized by appropriate conditional and prior probabilities and estimate the probability of error as well as the probability of receiving either a 1 or a 0. Start with a symmetrie binary communication channel characterized by the conditional probabiliti PLRlis 0.995 and POR I PR 0.003 The prior prebabilities of a 0 or a l are given by POs)-0.389 a PI ls 0.6 Xi...
help with number 4
4. Roll a die and flip a coin. Let Y be the value of the die. Let Z = 1 if the coin shows a head, and Z = 0 otherwise. Let X = Y + Z. Find the variance of X. 5. (a) If X is a Poisson random variable with = 3, find E(5*).
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