Evaluating Expectation and Variance of a Discrete Random Variable
Consider a discrete random variable \(X\) with probability mass function (pmf) given by \(P(X=x)\) for \(x\) in \(\{1, 2, 3, 4\}\). The probabilities are: – \(P(X=1)=0.1\) – \(P(X=2)=0.3\) – \(P(X=3)=0.4\) – \(P(X=4)=0.2\) (a) Calculate the expected value \(E(X)\). (b) Calculate the variance \(Var(X)\). (c) Interpret the standard deviation in the context of this distribution.
Unbiasedness of the Sample Mean
Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed random variables with mean \(\mu\). Show that the sample mean \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\) is an unbiased estimator for \(\mu\) by computing \(E(\bar{X})\). Explain your reasoning, ensuring it is applicable to both discrete and continuous random variables.
Evaluation of Unbiased Estimators in Discrete Distributions
Consider the following: 1. Let X be a random variable following a Poisson distribution with parameter \(\lambda\). An estimator for \(\lambda\) is given by \(T = X + c\), where c is a constant. Determine the value of \(c\) that makes \(T\) an unbiased estimator for \(\lambda\). 2. Let Y be a random variable following […]
Impact of Weighting Factor on Unbiasedness of Sample Means
Suppose a sample of random variables \(X_1, X_2, \dots, X_n\) have common mean \(\mu\) and variance \(\sigma^2\). A new estimator of \(\mu\) is defined by \(\hat{\mu} = \sum_{i=1}^n w_i X_i\), where \(w_i\) are weighting factors. Under what condition on the weights \(w_i\) is \(\hat{\mu}\) an unbiased estimator of \(\mu\)? Discuss the impact on unbiasedness if […]
Proof of Unbiasedness of the Sample Mean
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a distribution with expected value \(\mu\) and finite variance. Prove that the sample mean \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\) is an unbiased estimator of \(\mu\). Use either summation for discrete random variables or integrals for continuous random variables in your proof.
Determining Unbiasedness of a Linear Estimator
Let \(X\) be a random variable with expected value \(E(X)=\mu\). Consider the estimator \(T = aX\) for estimating \(\mu\), where \(a\) is a constant. (a) Determine the value of \(a\) such that \(T\) is an unbiased estimator of \(\mu\). (b) If \(a=0.9\), calculate the bias of the estimator \(T\) and comment on whether it is […]
Definition of an Unbiased Estimator
Define what it means for an estimator \(\hat{\theta}\) to be unbiased for a parameter \(\theta\).
Variance Transformation in Continuous Random Variables
Let X be a continuous random variable with probability density function \(f(x) = c(2-x)\) for \(0 \le x \le 2\) and \(f(x) = 0\) otherwise. (a) Determine the value of the constant \(c\) such that \(f(x)\) is a valid probability density function. (b) Calculate the expected value \(E(X)\) and the variance \(Var(X)\) of \(X\) by […]
Scaling Package Weights: A Linear Transformation Problem
A logistics company uses a scale that consistently overestimates package weights. To compensate, the weight measurement X (in kg) is transformed using the linear equation \(Y = 1.05X – 2\). Given that the mean measured weight is \(E(X)=50\) kg and the variance is \(Var(X)=16\) kg\(^2\), calculate the expected adjusted weight \(E(Y)\) and the variance \(Var(Y)\).
Expectation of a Linear Transformation of a Discrete Random Variable
Let X be a discrete random variable with the following probability mass function: – P(X = 1) = 0.2 – P(X = 2) = 0.5 – P(X = 3) = 0.3 Compute \(E(2X + 4)\).
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