Suppose $X$ follows an exponential distribution with parameter $\lambda > 0$. Fix $a > 0$. Define the random variable $Y$ by $Y = k, \text{ if } ka \leq X < (k+1)a, \quad k = 0, 1, 2, \dots$ Which of the following statements are correct?
The problem involves a random variable X following an exponential distribution and defines another discrete random variable Y based on intervals of X. We need to determine the properties and distribution of Y.
Defining Random Variable Y
The random variable X follows an exponential distribution with PDF $f_X(x) = \lambda e^{-\lambda x}$ for $x \geq 0$. The variable Y is defined as:
- $Y = k$, if $ka \leq X < (k+1)a$, for $k = 0, 1, 2, \dots$
This definition implies that Y is a discrete random variable taking non-negative integer values ($0, 1, 2, \dots$).
Evaluating Option A: $P(4 < Y < 5) = 0$
Since Y can only take integer values ($0, 1, 2, \dots$), it is impossible for Y to fall strictly between 4 and 5.
Therefore, the probability $P(4 < Y < 5)$ must be 0.
Conclusion: Option A is correct.
Evaluating Option B: Y follows an exponential distribution
An exponential distribution is a continuous probability distribution.
As established, the random variable Y is discrete.
A discrete random variable cannot follow a continuous distribution.
Conclusion: Option B is incorrect.
Evaluating Option C: Y follows a geometric distribution
We calculate the probability mass function (PMF) for Y.
For any integer $k \geq 0$:
$ P(Y=k) = P(ka \leq X < (k+1)a) $
$ P(Y=k) = \int_{ka}^{(k+1)a} \lambda e^{-\lambda x} dx $
$ P(Y=k) = \left[ -e^{-\lambda x} \right]_{ka}^{(k+1)a} $
$ P(Y=k) = -e^{-\lambda (k+1)a} - (-e^{-\lambda ka}) $
$ P(Y=k) = e^{-\lambda ka} - e^{-\lambda (k+1)a} $
Factor out $e^{-\lambda ka}$:
$ P(Y=k) = e^{-\lambda ka} (1 - e^{-\lambda a}) $
Let $p = 1 - e^{-\lambda a}$. Since $\lambda > 0$ and $a > 0$, we have $0 < e^{-\lambda a} < 1$, which implies $0 < p < 1$.
Also, let $q = e^{-\lambda a} = 1 - p$. Note that $0 < q < 1$.
The PMF becomes:
$ P(Y=k) = q^k \cdot p $
$ P(Y=k) = p \cdot (1-p)^k $
This is the probability mass function of a geometric distribution defined for $k = 0, 1, 2, \dots$, with parameter $p = 1 - e^{-\lambda a}$.
Conclusion: Option C is correct.
Evaluating Option D: Y follows a Poisson distribution
The PMF of a Poisson distribution is given by $P(Y=k) = \frac{e^{-\mu} \mu^k}{k!}$ for $k = 0, 1, 2, \dots$.
The derived PMF for Y is $P(Y=k) = (1 - e^{-\lambda a}) (e^{-\lambda a})^k$.
This form does not match the Poisson PMF (it lacks the $k!$ term and the structure is different).
Conclusion: Option D is incorrect.
Final Result
Based on the analysis, the correct statements are A and C.
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 $\bar{Y}$?
