Question
Let $X$ be a random variable with the cumulative distribution function $F_X(x) = \begin{cases} 0, & \text{if } x<0, \\ \frac{x+2}{5}, & \text{if } 0 \le x<2, \\ 1, & \text{if } x \ge 2. \end{cases}$ Then, which of the following statements are true?
To solve this problem, we are given the cumulative distribution function (CDF) \(F_X(x)\) of a random variable \(X\). We need to evaluate which of the given statements are true. Let's address each statement one by one.
- Statement: \(E(X^2) = \frac{4}{3}\)
- \(F_X(x) = 0\) if \(x < 0\)
- \(F_X(x) = \frac{x+2}{5}\) if \(0 \le x < 2\)
- \(F_X(x) = 1\) if \(x \ge 2\)
- Statement: \(P (|X - \frac{1}{2}| \ge 1) = \frac{19}{25}\)
- Statement: The upper bound of \(P(|X - \frac{1}{2}| \ge 1)\), using Chebyshev's inequality, is \(\frac{52}{75}\)
- Statement: The value of the moment generating function, \(M_X(t)\), at \(t = 1\) is \(\frac{e^3-1}{5}\)
After a careful step-by-step evaluation, the following statements from the problem are determined correct:
- The upper bound of \(P (|X - \frac{1}{2}| \ge 1)\), using Chebyshev's inequality, is given based on interpretations.
The calculations for \(E(X^2)\) showed neither was true.
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Important Questions from Random Variables
- A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function
$ f(x) = \begin{cases} e^{-x}, & \text{if } x>0, \\ 0, & \text{otherwise,} \end{cases} $
and that of Brand II follows a gamma distribution with the probability density function
$ g(x) = \begin{cases} \frac{x}{4} e^{-x/2}, & \text{if } x>0, \\ 0, & \text{otherwise.} \end{cases} $
The company uses the batteries of Brands I and II in proportion of $20\%$ and $80\%$ respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that $2$ years is - Consider a discrete random variable $X$ with the probability mass function
$ P(X = 0) = \frac{\theta}{3}, \ P(X = 1) = 1 - \frac{\theta}{2}, \ P(X = 2) = \frac{\theta}{6}, $
where $\theta \in (0,1)$ is an unknown parameter. In a random sample of size $90$ from this distribution, the observed counts for $X = 0, 1$ and $2$ are $20, 60$ and $10$, respectively. Then, the maximum likelihood estimate of $\theta$ is - Let $X$ be a random sample of size $1$ from the probability density function
$ f(x|\theta) = \begin{cases} \frac{3}{\theta^3} (\theta - x)^2, & \text{if } 0<x<\theta, \\ 0, & \text{otherwise.} \end{cases} $
If $ \left(\frac{X}{1-\lambda_1}, \frac{X}{1-\lambda_2}\right) $ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_i \in (0,1), \ i = 1,2, \ \lambda_1<\lambda_2$, and $\alpha \in (0,1)$, then which of the following statements is true? - Let $X_1, X_2, . . ., X_n$ be a random sample from a continuous distribution with the common probability density function
$ f(x|\theta) = \begin{cases} \frac{2\theta^2}{x^{\theta+1}}, & \text{if } x>2, \\ 0, & \text{otherwise,} \end{cases} $
where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y>\chi^2_{m,\beta}) = \beta$, where $Y \sim \chi^2_m$. For testing $H_0: \theta = 1$ against $H_1 : \theta>1$, a uniformly most powerful test of size $\alpha, \ 0<\alpha<1$, will reject $H_0$ if - Suppose we want to estimate the population mean $\bar{Y}$ of a variable for a finite population of size $85$, with $34$ Statisticians and $51$ Biologists. We consider the following sampling scheme:
A stratified random sample with $2$ strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where $12$ Statisticians and $15$ Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme.
Denote $\bar{y}_S, \bar{y}_B$, and $\bar{y}$ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of $\bar{Y}$?
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