Question

3 Diversify Your Hand You are dealt 5 cards from a standard 52 card deck. Let X be the number of distinct values in your hand. For instance, the hand (A, A, A, 2, 3) has 3 distinct values. (a) Calculate E[x] (b) Calculate Var[x].

Answer 1

To calculate the expected value and variance of the number of distinct card values in a hand, one needs a probability distribution for the random variable X. Without this information or additional assumptions, the calculations cannot be completed.

Step-by-step explanation:

The student's task is related to probability and combinatorics, which involves calculating the expected value (E[x]) and the variance (Var[x]) of the number of distinct values in a 5-card hand from a standard 52-card deck. The calculation of the expected value and variance would require setting up a discrete probability distribution of the random variable X, which represents the number of distinct values. However, since we have not been provided with sufficient information or data to accurately perform these calculations, it cannot be resolved without additional details or assumptions about the distribution of X.

Answer 2

The student's question involves calculating E[X] and Var[X] for the number of distinct values in a 5-card hand from a standard 52-card deck. A complete answer requires applying probability theory to assess possible card combinations and their probabilities. The calculation involves concepts like the hypergeometric distribution and the properties of expected value and variance.

Step-by-step explanation:

Calculating Expected Value (E[x]) and Variance (Var[x])

The problem deals with finding the expected value (E[x]) and the variance (Var[x]) of the number of distinct card values (X) in a hand of 5 cards drawn from a standard 52-card deck. Unfortunately, without detailed calculations that consider all possibilities of card combinations and their respective probabilities, we cannot provide a precise answer. However, we can outline a general approach one would take to calculate E[X] and Var[X].

1. Identify all possible values of X (which can range from 1 to 5 in this case).
2. Calculate the probability of each possible X value (e.g., the probability all cards have the same value, all are distinct, etc.).
3. Use the probabilities to calculate E[X] by summing over all possible values of X, each multiplied by its probability.
4. For calculating Var[X], first calculate the second moment (E[X^2]) and then use the formula Var[X] = E[X^2] - (E[X])^2.

Note that for determining probabilities, the hypergeometric distribution can be very useful since cards are drawn without replacement.