Probability And Statistics

To solve the problem, we need to compute the expected values (means) of the random variables and , which represent the number of blue and yellow balls drawn from the bag, respectively. We have a total of 9 balls: 5 blue and 4 yellow.

Step 1: Compute the Mean of

denotes the number of blue balls drawn. The expectation , or , can be calculated using the hypergeometric distribution formula:

$ n = 3 K = 5 N = 9 Y Y M = 4 7\bar{X} + 4\bar{Y} \bar{X} = \frac{5}{3} \bar{Y} = \frac{4}{3} 7\bar{X} + 4\bar{Y}$ is .

Let denote the number of defective items in the sample.

The mean is given by:

Calculate :

The variance is given by:

Now, calculate :

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Probability distribution:

To solve this problem, we want to find the probability that for natural numbers and . This is equivalent to finding the probability that .

1. Determine the possible values of :
The possible values of for any natural number are . Specifically, depends on :

2. Find when :
We have if and only if . This means that and must be opposite values. The possible pairs of opposite values are and .

3. Calculate the probability of :
We want to find the probability that and are not opposites. Consider the possible values of . For each value of , we determine the value of that makes .

• If , then . We need , which means .
• If , then . We need , which means .
• If , then . We need , which means .
• If , then . We need , which means .

In each case, there is exactly one value of that makes . Since there are 4 possible values for , the probability that is .

4. Calculate the probability of :
The probability that is .

Final Answer:
The probability that is .

Step 1: Define the random variable . represents the number of defective oranges drawn.

Step 2: Determine probabilities for .

Step 3: Calculate expected value .

Step 4: Calculate expected value .

Step 5: Calculate the variance .

To find the projection of on , first compute the vectors and using the given cross products. Then, use the projection formula: Substitute and into the formula.
Final Answer: .

To determine the probability of getting a sum of 4 or 5 when both dice are thrown, we must first understand the face distribution of the two dice:

• First Die: Two faces with 1, two faces with 2, one face with 3, and one face with 4.
• Second Die: One face with 1, two faces with 2, two faces with 3, and one face with 4.

Now, let's calculate the total number of possible outcomes when throwing both dice. Since each die has 6 faces, the total outcomes are:

Next, we calculate the number of favorable outcomes for each sum.

Sum of 4 Outcomes:

• (1,3) - First die shows 1, second die shows 3.
• (2,2) - First die shows 2, second die shows 2.
• (3,1) - First die shows 3, second die shows 1.

Now count the probability for each combination:

• Probability of (1,3):
• Probability of (2,2):
• Probability of (3,1):

Sum of 5 Outcomes:

• (1,4) - First die shows 1, second die shows 4.
• (2,3) - First die shows 2, second die shows 3.
• (3,2) - First die shows 3, second die shows 2.
• (4,1) - First die shows 4, second die shows 1.

Now count the probability for each combination:

• Probability of (1,4):
• Probability of (2,3):
• Probability of (3,2):
• Probability of (4,1):

Total Probability:

Add up all the probabilities of getting either a sum of 4 or 5:

Thus, the probability of getting a sum of 4 or 5 is . Therefore, the correct option is:

Step 1: Define the random variable . represents the number of defective oranges drawn.

Step 2: Determine probabilities for .

Step 3: Calculate expected value .

Step 4: Calculate expected value .

Step 5: Calculate the variance

Step 1: The given differential equation can be simplified and solved by separating variables and integrating. First, isolate and terms to obtain the relation between and .
Step 2: After applying the appropriate integration techniques, such as substitution and integration by parts, we get the general solution for .
Step 3: Use the initial condition to determine the constant of integration.
Step 4: Finally, substitute into the solution to get , which evaluates to . Thus, the correct answer is (1).

We are given the differential equation and the initial condition . To solve for , we first divide both sides of the equation by : We solve this first-order linear differential equation using the method of integrating factors. After solving, we substitute and calculate .
Final Answer: .

Concept:

• Mean
• Variance

Step 1: {Use the mean condition.} Mean , number of observations Sum of given numbers: Thus

Step 2: {Use the variance condition.} Variance Squares of known numbers: Thus

Step 3: {Find .}

Step 4: {Find and .} Since :

Step 5: {Form the required quadratic equation.} Roots: Sum of roots: Product: Thus equation: