This is for an Information Theory class. H(X) is entropy rate.
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This is for an Information Theory class. H(X) is entropy rate. Problem 8: Suppose that X is a random variable with a pr...
(10 points) Consider a discrete random variable X, which can only take on non-negative integer values,...
(10 points) Consider a discrete random variable X, which can only take on non-negative integer values, with E[Xk] = 0.8, k = 1,2, .... Use the moment generating function approach to find the pmf of Px(k), k = 0,1,....
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x <...
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which...
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which is symmetric about the value r 0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number. Hint: Use integration by parts
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p ,...
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
1.Let X be a random variable that takes on integer values 0 to 9 with equal...
1.Let X be a random variable that takes on integer values 0 to 9 with equal probability 1/10 a.Let Y-X mod(3); determine ly b. Let Y-6 mod(X + 1); determine Hy. For any non-negative integers, a and b, b# 0 by definition: a mod(b)-r For instance 27mod(12) 3 becaus2-+ k + where k and r are non-negative integers and 0 S r < b; 12
. (Markov’s Inequality) Let X be a non-negative random variable defined on the sample Ω i.e....
. (Markov’s Inequality) Let X be a non-negative random variable defined on the sample Ω i.e. X(s) ≥ 0 for all s ∈ Ω. Let a be some fixed positive number. (a) If you know nothing about the probability distribution of X, what can you say about P(X ≥ a)? (b) Now, if you know what the value of (E(X) = µ) is, can you say anything about P(X ≥ a)? Turns out something non-trivial can be said about this...
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 ...
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 , wherepEPr(X:-)-0.7 and Pr(X, =0).1-p=0.3. a) (10 pts.) Complete the table where k denotes the number of 1's in the n! sequence, andkkn-k b) (10 pts.) Calculate H(X) c) (10 pts.) Assume that Pr[T)]21-ε 0.9. Find the corresponding typical sequence set n) d) (10 pts.) Assume Pr[ 21-820.9. Find the corresponding smallest set B ). 2. Consider a random walk random variable X, on the graph in Figure 1....
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in h...
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
(10 points) Consider a discrete random variable X, which can only take on non-negative integer values, with E[Xk] = 0.8, k = 1,2, .... Use the moment generating function approach to find the pmf of Px(k), k = 0,1,....
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which is symmetric about the value r 0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number. Hint: Use integration by parts
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
1.Let X be a random variable that takes on integer values 0 to 9 with equal probability 1/10 a.Let Y-X mod(3); determine ly b. Let Y-6 mod(X + 1); determine Hy. For any non-negative integers, a and b, b# 0 by definition: a mod(b)-r For instance 27mod(12) 3 becaus2-+ k + where k and r are non-negative integers and 0 S r < b; 12
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 , wherepEPr(X:-)-0.7 and Pr(X, =0).1-p=0.3. a) (10 pts.) Complete the table where k denotes the number of 1's in the n! sequence, andkkn-k b) (10 pts.) Calculate H(X) c) (10 pts.) Assume that Pr[T)]21-ε 0.9. Find the corresponding typical sequence set n) d) (10 pts.) Assume Pr[ 21-820.9. Find the corresponding smallest set B ). 2. Consider a random walk random variable X, on the graph in Figure 1....
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
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