Standard Distributions

Step 1: Understanding the Concept:
This problem requires calculating the minimum sample size needed to estimate a population mean (the true mean reaction time) with a specified margin of error and confidence level. We assume the population is large enough to ignore the finite population correction.

Step 2: Key Formula or Approach:
The margin of error (E) in estimating a population mean is given by: We need to solve this formula for the sample size, .
Step 3: Detailed Explanation:
We are given the following values: - Confidence Level: 95%. The corresponding critical Z-value is . - Population Standard Deviation ( ): The estimate is given as 0.05 seconds. - Maximum Margin of Error (E): The error should not exceed 0.01 seconds, so . Substitute these values into the sample size formula: Since the sample size must be an integer, and we need to ensure the error does not exceed the specified amount, we must always round the result up to the next whole number. Therefore, the required sample size is . The question asks how large the sample should be, which means we need . The closest option that satisfies this condition is , although strictly is required. Given the options, 96 is the intended numerical answer. The format of the options in the OCR as is likely incorrect; it should be or . Based on our calculation, the required size is at least 96.04, so we choose the closest appropriate option.
Step 4: Final Answer:
A sample size of at least 97 is required. The closest option is .

Step 1: Understanding the Concept:
The question asks for the conditional expectation of Y given X=x, denoted . To find this, we first need to determine the conditional probability density function (PDF) of Y given X=x, denoted . Then, we can find the expectation of this conditional distribution.

Step 2: Key Formula or Approach:
1. Find the marginal PDF of X: . 2. Find the conditional PDF of Y given X: . 3. Calculate the conditional expectation: .

Step 3: Detailed Explanation:
1. Find the marginal PDF of X, : We can factor out the terms not involving y: The integral evaluates to: So, the marginal PDF is: (This means X follows an exponential distribution with rate 1). 2. Find the conditional PDF of Y given X, : This is the PDF of an exponential distribution with rate parameter . 3. Calculate the conditional expectation, : The expected value of an exponential distribution with rate parameter is . Since the conditional distribution of Y given X=x is exponential with rate , its expected value is:
Step 4: Final Answer:
The conditional expectation is .

Step 1: Understanding the Concept:
This problem involves finding a general formula for higher-order partial correlation coefficients when all the initial (zero-order) correlations are identical. We can find the pattern by recursively applying the formula for partial correlation.

Step 2: Key Formula or Approach:
The recursive formula for a partial correlation coefficient is: We will apply this starting from the first order and look for a pattern.

Step 3: Detailed Explanation:
Let all zero-order correlations be . First-order partial correlation (e.g., ): So, all first-order partial correlations are equal to . Second-order partial correlation (e.g., ): Using the recursive formula with the first-order partials: Substitute the value of : So, all second-order partial correlations are equal to . Third-order partial correlation (e.g., ): Using the same recursive logic, the third-order partial will be: Substitute the value of : The general formula for the -th order partial correlation coefficient is .
Step 4: Final Answer:
Every third order partial correlation coefficient is equal to .

Step 1: Understanding the Concept:
The covariance between two random variables X and Y is a measure of their joint variability. It is calculated using the formula Cov(X, Y) = E(XY) - E(X)E(Y). We need to compute the expectations E(X), E(Y), and E(XY) from the given joint PDF.

Step 2: Key Formula or Approach:
1. Calculate 2. Calculate 3. Calculate 4. Compute

Step 3: Detailed Explanation:
1. Calculate E(X): 2. Calculate E(Y): 3. Calculate E(XY): 4. Compute Cov(X,Y):
Step 4: Final Answer:
The covariance cov(X, Y) is .

Step 1: Understanding the Concept:
This problem involves a transformation of a discrete random variable. We need to find the probability of an event defined for the transformed variable Y. The first step is to translate the event for Y into an equivalent event for the original variable X. Then, we use the probability mass function (PMF) of X to calculate the required probability.

Step 2: Key Formula or Approach:
1. Convert the inequality for Y into an inequality for X: . 2. Solve for the possible integer values of X. 3. Calculate the probabilities for these values of X using the Binomial PMF: . 4. Sum the probabilities.

Step 3: Detailed Explanation:
1. Translate the event: Since X represents the number of successes, it must be a non-negative integer. The integer values of X that satisfy are . 2. The problem now is to calculate . 3. We use the Binomial PMF with and . The term becomes . - - - 4. Sum the probabilities: 5. Convert to a decimal: The calculated probability is 0.1445. This value is not exactly among the options. However, 0.165 is the closest option. It is likely there is a typo in the question or the options, but in an exam context, the closest answer would be chosen.
Step 4: Final Answer:
The calculated probability is . The closest option provided is 0.165.

Step 1: Understanding the Concept:
The multiple correlation coefficient, , measures the correlation between the variable and the best linear combination of the variables and . It quantifies how well can be predicted from and together.

Step 2: Key Formula or Approach:
The formula for the square of the multiple correlation coefficient is given in terms of the zero-order correlation coefficients:
Step 3: Detailed Explanation:
We are given the following correlation coefficients: - - - First, we calculate the squares of these coefficients: - - - Next, calculate the term : - Now, substitute these values into the formula for : Finally, take the square root to find :
Step 4: Final Answer:
The multiple correlation coefficient is approximately 0.856.

Step 1: Understanding the Concept:
We need to find the probability of the minimum of two i.i.d. discrete random variables being equal to a specific value. The key is to express the event in terms of events involving X and Y. A useful technique for the minimum is to work with the survival function, .

Step 2: Key Formula or Approach:
For with X and Y being independent: For discrete variables, the probability mass function can be found using the survival function: Or, equivalently, using the CDF:
Step 3: Detailed Explanation:
We want to find . Let's use the survival function approach. Since X, Y are i.i.d., . First, we need the probabilities for a single Poisson(1) variable, X. The PMF is . - - Now, calculate the required survival probabilities for X: - - Now, calculate the survival probabilities for Z: - - Finally, calculate : To match the options, we can write this with a common denominator of :
Step 4: Final Answer:
The probability P(Z=1) is .

Step 1: Understanding the Concept:
The given series is an infinite geometric series. The convergence or divergence of such a series depends entirely on the value of its common ratio, .

Step 2: Key Formula or Approach:
An infinite geometric series

Converges to the sum if the absolute value of the common ratio .
Diverges if . \end{itemize}
Step 3: Detailed Explanation:
The given series is .
The first term is , and the common ratio is .
We are also given the condition that .
Applying the convergence rule for a geometric series, the series converges if . Since , this simplifies to .
The series diverges if . Since , this simplifies to .
Therefore, the statement "It converges, if and diverges, if " is the correct description of the series' behavior.

Step 4: Final Answer:
The statement in option (B) correctly describes the conditions for convergence and divergence of the given geometric series.

Step 1: Understanding the Concept:
This problem can be modeled using a binomial distribution. Each child's gender is an independent trial. We are looking for the probability of a specific outcome (no girls) in a fixed number of trials (4 children). The number of families (800) is extra information not needed to calculate the percentage, but would be needed to find the number of families.

Step 2: Key Formula or Approach:
The binomial probability formula is given by: Where:
- is the number of trials (number of children in a family).
- is the number of successful outcomes (number of girls).
- is the probability of success on a single trial (probability of having a girl).
- is the probability of failure (probability of having a boy).

Step 3: Detailed Explanation:
Let's define the parameters for this problem:
- The number of trials, , is the number of children in each family, so .
- We are interested in families having no girls, so the number of successes, , is 0.
- The probability of having a girl, , is assumed to be 0.5.
- The probability of having a boy, , is also 0.5.
Now, we substitute these values into the binomial formula: To express this probability as a percentage, we multiply by 100:
Step 4: Final Answer:
The percentage of families having no girls is 6.25%.

Step 1: Understanding the Concept:
The question asks for the moment generating function (MGF) of a product of two independent random variables, one following a Beta distribution and the other a Gamma distribution. This requires knowledge of a specific identity in probability theory relating these distributions. The notation is ambiguous and could mean Gamma(shape, scale) or Gamma(shape, rate). However, a known theorem works if we interpret the parameters correctly.

Step 2: Key Formula or Approach:
There is a fundamental theorem in probability that states: If and are independent, then their product follows a Gamma distribution, specifically . The MGF of a Gamma distribution with shape parameter and rate parameter is .

Step 3: Detailed Explanation:
Let's analyze the given distributions: - is a Beta distribution with parameters and . - . The notation is ambiguous. For the theorem to apply, should be a Gamma distribution with shape . The provided notation is likely a typo for , i.e., shape= and rate= . Let's proceed with this assumption, as it leads to one of the given answers. Assuming . According to the theorem, with : If and are independent, then . Now, we find the MGF of , which follows a Gamma distribution with shape and rate . Using the MGF formula for a Gamma distribution: Substitute and : This result matches option (C). Any other interpretation of the notation for the Gamma distribution does not lead to a simple, standard result.
Step 4: Final Answer:
The moment generating function of Z is .