(a) Let X be a discrete random variable taking non-negative values; i.e. Rx = No = {0, 1, 2, ...}. Let a > 0. Show that E[X] > aP(X > a) and deduce that P(X > a) < E[X]/a, an inequality known as Markov's inequality. (b) Let X be a discrete random variable. By applying Markov's inequality to a well-chosen random variable, prove Chebyshev's inequality; that is, show that if a > 0 then Var[X] P (|X – EX|> a)< (c) Suppose that F ~ Bin(2m + 1, 5). Use Chebyshev's inequality to bound P[F > m]. From (c), deduce that the probability of a decoding error in the binary sym- metric channel (BSC) with symbol error probability p = can be made arbitrarily small by using a sufficiently long repetition code. сan

Q: X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and…

A: Moment generating function:

Q: The discrete random variable X has the probability distribution 2 4 P(X=x) k 2k 3k 4k 3.

A:

Q: (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If…

A: Given that, Z is discrete random variable. We know that,  E(z)= ∑Zi*p(zi) Where p(zi) is the…

Q: X10 be independent random variables with a comm bution over the interval (0, 1). Estimate P[Σ],X; <…

A:

Q: Q1: Prove (a) Markov's Inequality for non-negative continuous random variables U. (b) Chebeshey's…

A:

Q: Let X, Y are random variables (r.v.) and a, b, c, d are values. Complete the formulas using the…

A: Step 1: Step 2: Step 3: Step 4:

Q: The theorem which states that it's possible to show that a statistic is sufficient for the parameter…

A: Solution The given the expression if the factorization  theorem .

Q: Find the characteristic function of a uniformly distributed random variable X in the range [0,1] and…

A:

Q: Find fQ|X (q|x) for x ∈ {0, 1} and all q.

A: It is a fact of statistics. It is widely used.

Q: For a random variable X, suppose that E[X] = 2 and Var(X) = 4. Then (a) E[(4 + x)²] = (b) Var (5+5X)…

A: Given,E(X)=2Var(X)=4we know that…

Q: Let X1, X2, X3, X4 be independent random variables of size 4 from a gamma distribution G(a, X) : -…

A: We want to find the expected value E(Z)=X1+X2+X3+X4

Q: Let X be a Poisson random variable with A - :3 and Y be a Poisson random variable with X = 4. Assume…

A:

Q: O Find E[S„] and Var[S„].

A: It is given that Xis are iid and follows Uniform distribution with parameter a and b.

Q: c) Suppose that X and Y are jointly continuous random variable - x f(x,y) = {o* for 0< x <1 and 1< y…

A:

Q: Consider a continuous random variable X with pdf f,(x). If f,(x) Markov's inequality states that, =…

A:

Q: Suppose that X₁, X2, X3 are independent and identically distributed random variables with…

A: Solution

Q: (i) If X is a Poisson variable such that P(X=2) = 9P(X=6). Find the mean variance of X (ii) A…

A:

Q: where c is a constant that makes this a valid p.d.f.. Let Y = EX?. Use Markov's inequality to find…

A:

Q: (6) Let X be a Poisson random variable with parameter 9, then E(X²) = 1.

A: Given X~Pois(θ)

Q: Let (X1, ..., Xn) be a random sample from the Bernoulli(0) distribution. (a) Find the conditional…

A:

Q: Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are…

A: Sol:- (a) To show that X^2 and Y are independent, we need to show that for any sets A and B in the…

Q: Suppose the random variable y is a function of several independent random variables, say…

A: Given that the random variable y is a function of several independent random variables, say…

Q: Let x be a continuous random variable with P(X 3µ) < a by the Markov's inequality. What is (a) ?…

A:

Q: X is a discrete random variable with p.m.f given by P(X=x)= x/c where c is a constant. The support…

A:

Q: Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform…

A: Let X1, X2,.......Xn be the random sample having Uniform distribution (0,1)Y1=min{X1,…

Q: Let X1 and X2 be independent random variables with respective binomial distributions b(3,1/2) and…

A: The provided information is as follows:The  and  are two independent variables.The  follows binomial…

Q: TRUE OR FALSE. a.) Let X (X₁, X2,..., Xn)' be a random vector with joint cumulative distribution…

A:

Q: Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of…

A: To find the joint distribution of the random variables , , and, we can use the Jacobian method.…

Q: Suppose a random variable has a CDF given by Fx(x) = (1-e-x)u(x). Find the following quantities: (a)…

A: As per the Bartleby guildlines we have to solve first three subparts and rest can be reposted....…

Q: Let X be a Poisson random variable with mean λ = 20. Estimate the probability P(X ≥25) based on: (a)…

A: To estimate the probability P(X≥25) for a Poisson random variable X with mean λ=20, we will use four…

Q: Let X and Y be nonnegative integer-valued discrete random variables such that if 0 ≤ m ≤ n are…

A: Given the joint pmf of the nonnegative integer-valued discrete random variables X and Y:

Q: Let X and Y be independent Poisson random variables with parameters 0 and A, respectively. Thus the…

A:

Q: Suppose that X1, X2, X3 are independent and identically distributed random variables with…

A: # Given CDF of random variable x  F(x)=1-2^-x     : x&gt;0  let y=max(x1,X2,x3)   To find…

Q: Consider a random walk model: the random walk starts at 0, and each X_i has the same distribution…

A: From the given information, The random walk starts at 0. Each Xi has the same distribution such that…

Q: A simple example of a random variable is the indicator of an event A, which is denoted by IA: IA(W)…

A: Indicator function is defined as   IA(ω) =1 if ω ∈ A, 0 otherwise

Q: A random process is defined by X(t) = X₁ + Vt where Xo and V are statistically independent random…

A: Given-Mean of X0​: 2X01​+X02​​Mean of V: 2V1​+V2​​The mean of X(t)…

Q: Let X be  a continuous random variable with P(X<0)=0. When E(X)=\mu exists, P(X\ge 3\mu) \le…

A: