Random Variables - AP Statistics

By definition, a discrete random variable is a random variable whose values can be "counted" one by one. A continuous random variable is a random variable that can take any value on a certain interval. Of these choices, the number of lip products, the amount of money, and the number of midterms taken are all discrete random variables, as the respective values can be counted; however, the time taken to watch the first four seasons of a TV show is a continuous random variable, as not everyone will take the same amount of time to watch all those episodes (i.e. some might fastf-orward/replay parts of episodes).

The mean for any set of random variables is additive in the sense that

The difference is also additive, so we have

This means the mean of is .

The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers (even when negative), we have

.

So for this difference, we have

.

So the mean and variance are and , respectively. In addition to that, is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.

Because has a Poisson distribution we know that:

and

.

Therefore, since we are given the mean of 25, we can find its variance to also be 25.

No explanation available

No explanation available

The answer is 17.5. Simply take the values for each day, add them, and divide by the total number of days to obtain the mean:

We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean. First, multiply each possible outcome by the probability of that outcome occurring. Second, add these results together.

We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean.

First, multiply each possible outcome by the probability of that outcome occurring.

Second, add these results together.

First convert .

The player's three-point shooting follows a binomial distribution with and .

On average, he thus makes three-point shots per game.

This means he averages 12 points per game from three-point range if he tries to make 10 three-pointers per game.

Analyzing the data, there are more 6s than anything else (mode), the median is between and , the mean is , and the range is

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with and .

Second, we need to realize we can use a normal approximation of the binomial since and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be and our standard deviation will be , which is about .

There are four conditions that need to be satisfied for a binomial experiment:

1. Each trial must have two outcomes.

2. Each trial must be independent.

3. All trials must be identical.

4. The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of and a head is , so we multiply to the power of 4 (because we need 4 tails) by (for the single head).

So the probability is

.

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.

If the random variables are independent, the variances are additive in the sense that

.

So then the variance of the sum is

.

The standard deviation is the square root of the variance, so we have

.

This probability is simple to compute:

We want to add the probability that X is greater or equal to two. This means the probability that X=2 or X=3.

Adding the necessary probabilities we arrive at the solution.

The expected value is computed as

for any values of x that the random variable takes.

So we have