Let \(X\) be a random variable having binomial distribution \(B(7, p)\). If \(P(X = 3) = 5P(X = 4)\), then the sum of the mean and the variance of \(X\) is:
- \(\frac{105}{16}\)
- \(\frac{7}{16}\)
- \(\frac{77}{36}\)
- \(\frac{49}{16}\)
The Correct Option is C
Solution and Explanation
It is provided that,
\(P(X = 3) = 5P(X = 4) \;and \;n = 7\)
\(⇒ ^7C_3p^3q^4 = 5⋅^7C_4p^4q^3\)
\(⇒ q = 5p\) and also \(p + q = 1\)
\(⇒p=\frac{1}{6}\) and \(q=\frac{5}{6}\)
So, Mean =\(\frac{7}{6}\) and variance =\(\frac{35}{36}\)
Therefore, \(Mean + Variance\) = \(\frac{7}{6}+\frac{35}{36}\)
= \(\frac{77}{36}\)
Hence, the correct option is (C): \(\frac{77}{36}\)
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
Concepts Used:
Random Variable
A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's results. Random variables are often deputed by letters and can be classified as discrete, which are variables that have particular values, or continuous, which are variables that can have any values within a continuous range.
Random variables are often used in econometric or regression analysis to ascertain statistical relationships among one another.
Types of Random Variable
There are two types of random variables, such as:
- Discrete Random Variable - A discrete random variable can take only a finite number of unique values such as 0, 1, 2, 3, 4, … and so on. The probability distribution of a random variable has a list of probabilities differentiated with each of its possible values known as the probability mass function.
- Continuous Random Variable - An arithmetically valued variable is said to be continuous if, in any unit of measurement, whenever it can take on the values a and b. If the random variable X can presume an infinite and uncountable set of values, it is said to be a continuous random variable. When X takes any value in a given interval (a, b), it is also known to be a continuous random variable in that interval.