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): \(\frac{77}{36}\)
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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.