Suppose X follows a binomial distribution wit | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The probability mass function for a binomial distribution is given by $P(X=r) = { }^n C_r p^r q^{n-r}$, where $q = 1-p$. We are given the ratio $\

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(A) The probability mass function for a binomial distribution is given by $P(X=r) = { }^n C_r p^r q^{n-r}$, where $q = 1-p$. We are given the ratio $\frac{P(X=r)}{P(X=n-r)}$. Substituting the formula: $\frac{P(X=r)}{P(X=n-r)} = \frac{{ }^n C_r p^r q^{n-r}}{{ }^n C_{n-r} p^{n-r} q^r}$. Since ${ }^n C_r = { }^n C_{n-r}$, the expression simplifies to: $\frac{P(X=r)}{P(X=n-r)} = \frac{p^r q^{n-r}}{p^{n-r} q^r} = \left(\frac{p}{q}\right)^{r - (n-r)} = \left(\frac{p}{q}\right)^{2r-n} = \left(\frac{q}{p}\right)^{n-2r}$. For this expression to be independent of $n$, the base must be $1$ (or the exponent must be $0$, but it must hold for all $r$). Thus, $\frac{q}{p} = 1$, which implies $q = p$. Since $p + q = 1$, we have $p + p = 1$, which gives $2p = 1$. Therefore, $p = \frac{1}{2}$.

Suppose $X$ follows a binomial distribution with parameters $n$ and $p$, where $0 < p < 1$. If $\frac{P(X=r)}{P(X=n-r)}$ is independent of $n$ for every $r$, then $p$ is equal to

Similar Questions

In the Compton scattering process,the incident $X$-radiation is scattered at an angle of $60^{\circ}$. The wavelength of the scattered radiation is $0.22 \ \mathring{A}$. The wavelength of the incident $X$-radiation in $\mathring{A}$ units is:

Let $A = \{1, 2, \{3, 4\}, 5\}$. Which of the following statements is incorrect and why?
${3, 4} \subset A$

The $V-I$ graph for a conductor at temperatures $T_1$ and $T_2$ is as shown in the figure. Then $(T_2 - T_1)$ is proportional to

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is

The eccentricity of the ellipse $9x^2 + 5y^2 - 18x - 20y - 16 = 0$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module