Defining Random Variables and Probability Distributions - Statistics and Probability

1. Which graph is most appropriate for a discrete probability distribution: histogram or bar graph?

   Bar graph (separate bars for each value of $X$). Discrete values require separated bars, unlike continuous histograms.

2. What is meant by the event $\{X\le 2\}$ in terms of outcomes?

   All outcomes whose assigned value satisfies $X\le 2$. Includes outcomes where $X$ equals 0, 1, or 2.

3. What is $P(X=1)$ if $X$ is the number of heads in $2$ fair coin flips?

   $\frac{1}{2}$. HT and TH each give 1 head, so 2 out of 4 outcomes.

4. What should the vertical axis represent when graphing a discrete distribution for $X$?

   The probability $P(X=x)$. Heights show likelihood of each outcome.

5. Identify all possible values of $X$: roll two dice; $X$ is the sum. What values can $X$ take?

   $X\in\{2,3,4,5,6,7,8,9,10,11,12\}$. Minimum sum is 1+1=2, maximum is 6+6=12.

6. Which display is appropriate to show a probability distribution table for $X$?

   A two-column table of $x$ and $P(X=x)$. Lists each outcome with its probability.

7. What is the probability distribution of a discrete random variable $X$?

   A list/table of each $x$ value with its probability $P(X=x)$. Shows how probability is distributed across values.

8. Identify the random variable: flip a coin; let $X$ be the number of heads. What is $X$?

   $X\in\{0,1\}$. Counts heads: 0 for tails, 1 for heads.

9. Which notation represents the probability distribution of a discrete random variable $X$?

   $P(X=x)$ for each possible value $x$. Shows the probability that $X$ takes each specific value.

10. What graph is most appropriate for a discrete probability distribution of $X$?

    A probability histogram (bar graph) with heights $P(X=x)$. Bars show probabilities for each discrete value.

11. Identify the random variable: flip $2$ coins; $X$ is number of heads. What are possible $X$ values?

    $\{0,1,2\}$. Can get 0, 1, or 2 heads when flipping two coins.

12. Identify whether this is a valid distribution: $P(X=1)=0.6$, $P(X=2)=0.5$.

    Invalid because $0.6+0.5=1.1\ne 1$. Probabilities must sum to exactly 1, not exceed it.

13. What is the sample space for an experiment?

    The set of all possible outcomes of the experiment. Contains every result that could occur when performing the experiment.

14. What is $P(X=7)$ if $X$ is the sum of two fair dice?

    $\frac{6}{36}$. Six ways to get 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

15. Find the missing probability: $P(X=0)=0.2$, $P(X=1)=0.5$, $P(X=2)=?$ and totals $1$.

    $P(X=2)=0.3$. Use $1-0.2-0.5=0.3$ since probabilities sum to 1.

16. If $X$ is the sum of two dice, what are the minimum and maximum possible values of $X$?

    Minimum $2$, maximum $12$. Smallest sum is $1+1=2$, largest is $6+6=12$.

17. Which notation represents the probability that $X$ equals $3$?

    $P(X=3)$. Standard notation for probability at a specific value.

18. Which display best matches a probability distribution table: list of $x$ and $P(X=x)$ or raw outcomes?

    A table listing each $x$ with its probability $P(X=x)$. Shows the probability distribution as value-probability pairs.

19. What is the sample space for a probability experiment?

    The set of all possible outcomes of the experiment. Contains every result that could occur.

20. Identify $X$ values: choose a person; $X$ is number of siblings. What type of variable is $X$?

    Discrete random variable. Takes countable values (0, 1, 2, ...).

21. What two conditions must a discrete distribution satisfy for $X$?

    $0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Each probability is valid and they exhaust all possibilities.

22. What two conditions must a discrete probability distribution satisfy?

    $0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Probabilities must be between 0 and 1, and sum to exactly 1.

23. Find $P(X=7)$ for two fair dice where $X$ is the sum.

    $\frac{6}{36}$. Six ways to get 7 out of 36 total outcomes.

24. What is the standard horizontal axis label when graphing a probability distribution for $X$?

    Possible values of the random variable $X$. Shows which values the random variable can take.

25. What does it mean to define $X$ as a random variable for a quantity of interest?

    Assign a number $X()$ to each outcome $$ to represent that quantity. Creates a numerical representation of the quantity for each possible outcome.

26. Compute $P(X\le 2)$ for one fair die roll where $X$ is the outcome.

    $\frac{2}{6}$. Two favorable outcomes out of six possible.

27. What should the horizontal axis represent when graphing a discrete distribution for $X$?

    The possible values $x$ of the random variable $X$. Shows which outcomes are being measured.

28. What is a random variable in probability?

    A function assigning a number to each outcome in a sample space. Maps outcomes to numbers for mathematical analysis.

29. What is the key difference between a data histogram and a probability histogram for $X$?

    Data uses frequency; probability uses relative frequency $P(X=x)$. Data shows counts; probability shows proportions.

30. Identify the event: roll one die; let $X$ be the outcome. What event corresponds to $X\le 2$?

    $\{1,2\}$. Outcomes 1 and 2 satisfy $X\le 2$.

31. Identify the random variable: roll a die; $X$ is the number shown. What are possible $X$ values?

    $\{1,2,3,4,5,6\}$. A standard die shows integers from 1 to 6.

32. Find the missing probability: $P(X=0)=0.2$, $P(X=1)=0.5$. What is $P(X=2)$?

    $0.3$. Probabilities must sum to 1: $0.2 + 0.5 + ? = 1$.

33. Find $P(X=2)$ if $X$ is the outcome of one fair die roll.

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

34. Which bar graph feature indicates an error when graphing a probability distribution for $X$?

    Bar heights (probabilities) do not sum to $1$. Valid distributions require all probabilities to sum to exactly 1.

35. What is the standard vertical axis label when graphing a probability distribution for $X$?

    Probability $P(X=x)$. Shows the likelihood of each value occurring.

36. What is $P(X=2)$ if $X$ is the number of heads in $2$ fair coin flips?

    $\frac{1}{4}$. Only HH gives 2 heads out of 4 equally likely outcomes: HH, HT, TH, TT.

37. Identify the sample space: roll one fair die once. What is $S$?

    $S=\{1,2,3,4,5,6\}$. Die shows 1 through 6 as possible outcomes.

38. What is a random variable in probability?

    A function assigning a numerical value to each outcome in a sample space. Maps experimental outcomes to numbers for mathematical analysis.

39. Find and correct the error: a distribution lists $P(X=1)=0.6$ and $P(X=2)=0.5$.

    Invalid because $0.6+0.5\ne 1$; probabilities must sum to $1$. Probabilities exceed 1 when summed.

40. What is the probability mass function (pmf) for a discrete random variable $X$?

    The rule listing each $x$ with its probability $P(X=x)$. Specifies the probability for each discrete value $X$ can take.

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