Random Variables and Probability Distributions - AP Statistics

1. State the formula for the binomial probability.

   $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combines combinations with probability.

2. Determine the probability of drawing a red card from a standard deck.

   $\frac{1}{2}$. 26 red cards in 52 total.

3. What is the probability of getting a sum of 7 when rolling two dice?

   $\frac{1}{6}$. 6 ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

4. Which property must a PDF satisfy?

   The total area under the curve is 1. Area represents total probability.

5. What is the Poisson distribution?

   A distribution representing the number of events in a fixed interval of time or space. Rate parameter $\lambda$ is key.

6. Find the variance of a fair six-sided die roll.

   $\frac{35}{12}$. $E(X^2) - [E(X)]^2 = \frac{35}{12}$

7. Calculate the probability of drawing an ace from a standard deck of cards.

   $\frac{1}{13}$. 4 aces in 52 cards.

8. Find the probability of rolling a 3 on a fair six-sided die.

   $\frac{1}{6}$. Each outcome equally likely.

9. What is a random variable?

   A variable whose values depend on outcomes of a random phenomenon. Its value is determined by chance events.

10. What is the probability density function (PDF)?

    A function that describes the likelihood of a continuous random variable to take on a value. For continuous variables; area gives probability.

11. Define a continuous random variable.

    A random variable with an infinite number of possible values in an interval. Can take any value within a range.

12. What does it mean if a random variable is normally distributed?

    It follows a bell-shaped curve, symmetric about the mean. 68-95-99.7 rule applies.

13. Calculate the expected value of a fair six-sided die roll.

    $3.5$. $\frac{1+2+3+4+5+6}{6} = 3.5$

14. What is the expected value of a uniform distribution over $[a, b]$?

    The midpoint, $\frac{a+b}{2}$. Average of the endpoints.

15. What is a probability distribution?

    A function that gives the probabilities of different outcomes of a random variable. Maps outcomes to their probabilities.

16. What is the standard deviation of a standard normal distribution?

    1. Standard normal is $N(0,1)$.

17. What is the mean of a standard normal distribution?

    0. Standard normal is $N(0,1)$.

18. Identify the property of a probability distribution function.

    The sum of all probabilities is 1. All probabilities must add to 1.

19. What is the probability of success in a Bernoulli trial?

    Denoted as $p$, where $0 \text{<} p \text{<} 1$. Complement is $(1-p)$.

20. What is the cumulative distribution function (CDF)?

    A function giving the probability that a random variable is less than or equal to a value. Cumulative probability up to a value.

21. What is the main characteristic of a geometric distribution?

    Counts the number of trials until the first success. Memoryless property applies.

22. Determine the probability of drawing a heart from a standard deck of cards.

    $\frac{1}{4}$. 13 hearts in 52 cards.

23. What is a binomial distribution?

    The distribution of the number of successes in a fixed number of independent Bernoulli trials. Fixed trials with constant success probability.

24. Identify the type of random variable: number of heads in 10 coin flips.

    Discrete random variable. Countable outcomes (0-10 heads).

25. Identify the main difference between binomial and hypergeometric distributions.

    Binomial involves replacement; hypergeometric does not. Replacement changes the probability.

26. What is the hypergeometric distribution?

    Distribution of successes in draws without replacement from a finite population. Population size affects probability.

27. Find the probability of getting at least one 6 in two dice rolls.

    $\frac{11}{36}$. $1 - P(\text{no sixes}) = 1 - (\frac{5}{6})^2$

28. Find the probability of no heads in 3 coin flips.

    $\frac{1}{8}$. $(\frac{1}{2})^3 = \frac{1}{8}$

29. Identify the type of random variable: height of students in a class.

    Continuous random variable. Can take any value in range.

30. Define a Bernoulli random variable.

    A random variable with two possible outcomes: success or failure. Simplest discrete distribution.

31. Define a discrete random variable.

    A random variable with a countable number of possible values. Can take values like 1, 2, 3, etc.

32. What is the mean of a probability distribution?

    The expected value of the distribution. The central tendency of the distribution.

33. Identify the key feature of a uniform distribution.

    All outcomes are equally likely within the range. Constant probability density function.

34. Define the probability mass function (PMF).

    A function giving the probability that a discrete random variable is exactly equal to some value. For discrete variables only.

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