Question

Suppose that r chips are drawn with replacement from an urn containing n chips, numbered 1 through n. Let V denote the sum of the numbers drawn. Find E(V ).

Answer 1

The expected value of the sum V of r chips drawn with replacement from an urn with n chips is the product of r and the mean of the chip numbers, yielding r*(n+1)/2.

Step-by-step explanation:

The student's question involves finding the expected value (E(V)) of the sum of r chips drawn with replacement from an urn containing n uniquely numbered chips. To solve this, we can use the formula E(X) = μ = Σ xP(x), where x is a value of the random variable, and P(x) is the probability of x occurring. The expected value of each draw is the mean of the numbers on the n chips, which is (n+1)/2. Since each of the r draws is independent and identically distributed, the overall expected value is r times the expected value of a single draw which is r*(n+1)/2.

Answer 2

To find the expected value, E(V), of the sum V of chips drawn with replacement from an urn containing n chips numbered 1 through n, we need to multiply each possible value of the sum by its probability and add the products. The formula for E(V) is E(V) = Σ (V * P(V)), where Σ denotes the sum and P(V) is the probability of getting V. An example calculation is provided for drawing chips with replacement from an urn containing 3 red chips and 8 blue chips.

Step-by-step explanation:

The expected value, denoted as E(V), is the mean of a discrete random variable V. In this case, we are drawing chips with replacement from an urn containing n chips. Each chip has a number from 1 to n. The sum of the numbers drawn, V, is a random variable. To find E(V), we need to multiply each possible value of V by its probability and add the products. The formula for E(V) is given as E(V) = Σ (V * P(V)), where Σ denotes the sum and P(V) is the probability of getting V.

For example, let's consider a specific case where we have an urn with 3 red chips and 8 blue chips. If we draw 2 chips with replacement, the possible values of V are 2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 16. To find E(V), we need to calculate the probability of getting each of these values and multiply it by the value itself. Then we sum up these products to get the expected value.

For instance, the probability of getting V = 7 is the probability of drawing a red chip followed by a blue chip, which is (3/11) * (8/11) = 24/121. Multiply this probability by 7 to get the contribution of V = 7 to the expected value. Repeat this calculation for all possible values of V and then sum up the contributions to get E(V).