Derive formulas for the mean and variance of a Poisson random variable with parameter λ using its MGF.

Q: Suppose that Y, Y2.,Yn parameter (2,8+ 4). are independent random variables from a gamma…

A:

Q: Let us consider a discrete random variable having the pmf given by, PX(k) = { (1/3) , k = 1   (2/3)…

A: The moment generating function of x is given by: Mxs=EesX=13es+23e2s+0e3sMxs=13es+23e2s

Q: Let X1, X2,..Xn be a random sample from a Poisson distribution with variance 0. Find the UMVUE for…

A: Let x1, x2, …, xn be random sample from poisson distribution with variance θ. The probability mass…

Q: 2. Let us consider a discrete random variable having the pmf given by, k = 1 Px (k) = {2 k = 2. 0, k…

A: Calculate the moment generating function:

Q: Consider a standard model with a random sample of n observations from a population with a proportion…

A: The given test is two proportions test.

Q: The joint pdf of X and Y is (i) ||-|x|-° fx.x (x, y) =-e Are X and Y statistically independent…

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Q: Two iid Gaussian random variables X, and X, have mean my and variance . Their sample mean and sample…

A: Given information: Given that X1 and X2 are two iid Gaussian random variables with mean mx and…

Q: Two random variables X and Y are known to be uncorrelated. When considered separately, X is known to…

A: If two random variable X and Y are uncorrelated then Cov(X,Y)= 0 I.e  X and Y are independent. If X…

Q: the moment generating function of the random variable is unic true false

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Q: For binomial random variable X with n=10, p=0.3, plot its pdf and cdf(cumulative distribution…

A: Draw a random sample of size 2000 with n=10 and p=0.3 from binomial distribution, Then new window…

Q: Show that the random variable X = Z and Y = 2?. fact that they are statistically dependent. are…

A: Answer: Using the given data,

Q: 2. Let us consider a discrete random variable having the pmf given by, k = 1 Px(k) = {2 k = 2. 3' 0,…

A: We have been given the pmf as,   Px(k)=13,      k=123,      k=20,        k=3 The mgf is given by,…

Q: Let X and Y be independent Poisson random variables with parameters 0.25 and 0.35,

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

Q: A certain type of tree has seedlings randomly dispersed in a large area, with the mean density of…

A: The number of events occur within the given interval is calculated by poisson distribution. The…

Q: Derive the likelihood function for an causal and invertible AR(2) Gaussian process and find the…

A: please refer the answer below:

Q: Explain the Covariance functions as applicable to random processes.

A: The autocovariance function CX(s,t) of a random process {X(t)} is a function of two times s and t.…

Q: The characteristic function for a Gaussian random variable X, having a mean value of zero, is ¢, (@)…

A:

Q: A survey found that 0.05% of the people in a city are infected by the COVID virus. If we randomly…

A: Given that , probability, p= 0.05% = 0.0005 n= 5000 So,  X~ binomial (5000,0.0005) Hence, mean = np…

Q: The normal random variable with mean = 0 and standard deviation = 1 is said to have the…

A: If X is random variable from standard distribution with mean(mu) = 0 and standard deviation (sigma)…

Q: Let X1,..., Xn be iid random variables from Poisson(A). Find a sufficient statistic for X. Use this…

A: Given information: The probability density function of the random variable X is as given below:

Q: Consider a random sample X1, X2, ....., X30 of size 30, drawn from a  normal distribution with µ=…

A: Consider a random sample X1, X2, ….., X30 of size 30, drawn from a normal distribution with mean (μ)…

Q: Let be a geomerric random variable with prf y=1,2,3... where 9 Find. estimator using matimum…

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Q: Develop a random variate generator for a random variable X with the pdf if 0 <x< 1, 2 f(x) = 3 if 1…

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Q: Suppose that the number of customer that enter a bank in an hour is a Poisson random variable, and…

A: Let X be poison random variable P(X=0)=0.05 variance of X=?

Q: Let X1,X2,...Xn be a random sample from a Poisson population with parameter A. Then. the sample mean…

A: The Pmf of the Poisson distribution is given below: fx=e-λλxx!;x=0,1, 2, ...,n The maximum…

Q: Calculate the mean and variance of this distribution using the MGF obtained in (a).

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Q: If X and Y are independent chi-square random variables with respective parameters r and r,, show…

A: Answer:

Q: Compute the mean and variance of a random variable whose distribution is uniform on the interval [a,…

A: Given: Consider a random variable X following Uniform distribution with the range a and b. The…

Q: If the random sample of X1,…, Xn comes from the uniform (0, Ɵ) distribution, which of the following…

A: Given information: Given that the independent random variables X1, X2, … , Xn are uniformly…

Q: Suppose the random variable X follows an Inverse Gaussian distribution with parameters u and 0. Use…

A: From the given information, the random variable X follows an Inverse Gaussian distribution with…

Q: Let X1,..., Xn be a random sample from the Pareto distribution given above. (i) Find the Method of…

A: Solution

Q: Derive the variance of a Poisson random variable with parameter λ. [Hint. Start with the…

A:

Q: If a random variable X follows the process below, dX; = -kX;dt + odWt, W is a standard Brownian…

A: A random variable is a numerical description of the outcome of a statistical experiment. A random…

Q: A Gamma random variable with expected value n and variance equal 2n, where n is a natural number,…

A: The parameters are estimated as follows: EX=αβ=nVX=αβ2=2nβ=VXEX=2nn=2α=n2

Q: Suppose that Y, Y2,.-.,Y, are independent random variables from a gamma distribution of the…

A:

Q: Find the characteristic function of a random variable having the binomial distribution with…

A: The characteristic function of a random variable having the binomial distributio with parameters p…

Q: Derive the moment generating function (MGF) of a random variable ?.

A: Given that      We have to derive the moment generating function of Y

Q: Let X be Bernoulli random variable, find the moment generating function X and find its mean,…

A: The answer is given using the concept of the Bernoulli random variable.

Q: Two random variables X and Y are distributed with an unknown joint PDF. From simultaneous…

A: Given information: The scatterplot of the simultaneous measurements of two random variables X and Y…

Q: Let X be a discrete random variable with expectation -0.5 and variance 3. Let Y=−2X+8, find the…

A: The expectation and variance of discrete random variable X are -0.5 and 3, respectively.

Q: Suppose the random variable X follows an Inverse Gaussian distribution with parameters u and 8. Use…

A: From the given information, the random variable X follows an Inverse Gaussian distribution with…

Q: Calculate the variance of a random variable X whose characteristic function is (1+ e3it)² f(t) = 4

A: We have given that the characteristics function of random variable X and we have to calculate the…

Q: A survey found that 0.07% of the people in a city are infected by the COVID virus. If we randomly…

A:

Q: • The pdf of a uniform random variable is given by: Ax) = -0,xb • Find the CDF.

A: Given that, X ~ Ua, b fx=1b-a,a≤x≤b0,x&lt;a and x&gt;b We know that, the cumulative distribution…

Q: Let X and Y denote two discrete random variables taking the values in x and y respectively. The…

A:

Q: X is an poisson random variable with parameter λ = 4 Calculate P {X ≤ 3}

A: It is given that X is the random variable and the mean given by λ is equal to 4 The probability of X…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.