Binomial Distribution

Step 1: Understanding the Given Expansion
The given function is: The coefficient of in the expansion of this expression is denoted as , meaning:

Step 2: Finding
To determine , we evaluate the function at :

Step 3: Computing
By differentiation, Setting , Substituting into : Thus,

Step 1: Ensure that the probabilities sum to 1. The total probability must sum to 1, so we write the equation: Simplify the equation: Solve this quadratic equation: Using the quadratic formula: Thus, we have two possible values for : Since must be non-negative, we take .

Step 2: Calculate the expected value . The expected value is calculated as: Substitute :

Step 3: Calculate the expected value of , i.e., . The expected value is calculated as: Substitute :

Step 4: Calculate the variance . The variance is given by: Substitute the values and : Thus, the corrected variance is .

Step 1: Understanding Assertion (A)
The given series: is a known series expansion which simplifies to: This means that Assertion (A) is correct.

Step 2: Understanding Reason (R)
The Reason (R) states: This is the binomial expansion for , which is a standard result in mathematical series. Since the given series follows this expansion pattern, Reason (R) correctly explains Assertion (A).

Step 3: Conclusion
Since both Assertion (A) and Reason (R) are correct and (R) provides a valid explanation for (A), the correct answer is:

We start with the given equation:

Step 1:
Find a common denominator for the right-hand side: Expanding the numerators: Adding these, Grouping like terms: Equating with the left-hand side: Since there is no , , and terms on the left-hand side, we set: Solving these equations, we get: From and substituting the values, we find: Thus, the correct answer is .

Step 1: Use the given mean and variance of the binomial distribution. Given that the mean and variance , for a binomial distribution, we know the following relationships: where is the number of trials and is the probability of success in each trial. From the mean , we have: From the variance , we have:

Step 2: Solve for and . To solve for and , substitute into the variance equation: Simplify the equation: Multiply through by 25 to clear the denominators: Substitute into to solve for : Thus, and .

Step 3: Calculate . We are given . First, we calculate using the complement rule: The probability of is given by: Substitute : Thus: Now, equate this to : Simplifying:

Step 4: Calculate . We have , so: Thus, the correct value of is .

We are given that the random variable follows a binomial distribution with the following properties: The mean is . The probability .

Step 1: Determine using the Mean
Since the mean of a binomial distribution is , we have:

Step 2: Use
For a binomial distribution, the probability of exactly 2 successes is: Substitute and the given : Using the binomial coefficient , the equation becomes: Simplify to obtain:

Step 3: Solve for
Due to the complexity of the expression, we test possible values for . By substituting , we find that: confirming that satisfies the condition.

Step 4: Compute the Variance
The variance of a binomial distribution is given by: Using and (since ), we have: Thus, the variance of is: