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Show that the following pair of statements are logically equivalent with proper explanation:
1. ∀x (P(x) ∧ Q(x)) and ∀x P(x) ∧ ∀x Q(x)
2. ¬ (∀x P(x)) and ∃x ¬P(x)
3. ¬ (∃x P(x)) and ∀x ¬P(x)
6. Determine whether each of the following statements about Fibonacci numbers is true or false. Note
The first 10 terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.
a. If n is even, then F is an odd number.
b. 2F-Fn-2 = Fn+1 for n 23
A certain type of storage battery lasts, on average, 3.0 years with a standard deviation of 0.5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last less than 2.3 years.
The meat department at a local supermarket specifically prepares its "1-pound" packages of ground beef so that there will be a variety of weights, some slightly more and some slightly less than 1 pound. Suppose that the weights of these "1-pound" packages are normally distributed with a standard deviation of 0.15 pound. What is the probability that a randomly selected package of ground beef will weigh less than 0.80 pound?
GIVEN THE EQUATION
4x²-3x + 4 = y³-x³ / 2
FIND ALL POSITIVE INTEGER SOLUTIONS OF x and y.
A kinder teacher developed a coloring worksheet for her pupils. Using this working sheets, the pupil's performance has a mean score of 90 and a standard deviation of 10. Fifty kinder students from a certain barangay were asked to answer the said worksheet and found that their mean score was 95 with a standard deviation of 5. Test the hypothesis at 1% significance level.
1. Construct the sample space of an experiment of tossing three unbiased coins. Determine its discrete probability distribution.
2. Construct a discrete probability distribution for a basketball team's probability of winning in 4 consecutive games.
Given the population of numbers 3,6,8,9 and 4. Suppose samples of size 3 are drawn from this population.
Construct the sampling distribution of sample means.
What is the probability P(X > 4)? (CLUE: Add all the probabilities for the values of the random variable greater than 4
Given the population of numbers 3,6,8,9 and 4.Suppose samples of a size 3 are drawn from this population.
1. What is the mean and the variance of population?
2. how many different samples of size 3 can be drawn from the population? List them with their corresponding mean.
