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Exam 2: Review of Probability

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Question 1

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The cumulative probability distribution shows the probability

A) that a random variable is less than or equal to a particular value.
B) of two or more events occurring at once.
C) of all possible events occurring.
D) that a random variable takes on a particular value given that another event has happened.

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Question 2

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Using the fact that the standardized variable Z is a linear transformation of the normally distributed random variable Y, derive the expected value and variance of Z.

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Think of the situation of rolling two dice and let M denote the sum of the number of dots on the two dice.(So M is a number between 1 and 12.) (a)In a table, list all of the possible outcomes for the random variable M together with its probability distribution and cumulative probability distribution.Sketch both distributions.

Math and verbal SAT scores are each distributed normally with N(500,10000). (a) What fraction of students scores above 750? Above 600? Between 420 and 530? Below 480? Above 530?

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that $\operatorname { corr } ( X , Y ) = \operatorname { corr } \left( X ^ { * } , Y ^ { * } \right) \text {, }$ where $X ^ { * } = a + b X \text { and } Y ^ { * } = c + d Y \text {, }$ and where a, b, c , and d are arbitrary non-zero constants.

The kurtosis of a distribution is defined as follows: a. $\frac { E \left[ \left( Y - \mu _ { Y } \right) ^ { 4 } \right) } { \sigma _ { Y } ^ { 4 } }$ b. $\frac { E \left[ \left( Y ^ { 4 } - \mu _ { Y } ^ { 4 } \right) \right) } { \sigma _ { Y } ^ { 2 } }$ c. $\frac { \text { skewness } } { \operatorname { var } ( Y ) }$ d. $E \left[ \left( Y - \mu _ { Y } \right) ^ { 4 } \right)$

The height of male students at your college/university is normally distributed with a mean of 70 inches and a standard deviation of 3.5 inches.If you had a list of telephone numbers for male students for the purpose of conducting a survey, what would be the probability of randomly calling one of these students whose height is (a)taller than 6'0"? (b)between 5'3" and 6'5"? (c)shorter than 5'7", the mean height of female students? (d)shorter than 5'0"? (e)taller than Shaq O'Neal, the center of the Miami Heat, who is 7'1" tall? Compare this to the probability of a woman being pregnant for 10 months (300 days), where days of pregnancy is normally distributed with a mean of 266 days and a standard deviation of 16 days.

Use the definition for the conditional distribution of Y given X=x and the marginal distribution of X to derive the formula for Pr(X=x, Y=y) . This is called the multiplication rule. Use it to derive the probability for drawing two aces randomly from a deck of cards (no joker), where you do not replace the card after the first draw. Next, generalizing the multiplication rule and assuming independence, find the probability of having four girls in a family with four children.

For a normal distribution, the skewness and kurtosis measures are as follows:

Consistency for the sample average $\bar { Y }$ can be defined as follows, with the exception of

The correlation between X and Y a. cannot be negative since variances are always positive. b. is the covariance squared. c. can be calculated by dividing the covariance between $X$ and $Y$ by the product of the two standard deviations. d. is given by $\operatorname { corr } ( X , Y ) = \frac { \operatorname { cov } ( X , Y ) } { \operatorname { var } ( X ) \operatorname { var } ( Y ) }$ .

The textbook formula for the variance of the discrete random variable Y is given as $\sigma _ { Y } ^ { 2 } = \sum _ { i = 1 } ^ { k } \left( y _ { i } - \mu _ { Y } \right) ^ { 2 } p _ { i } .$ Another commonly used formulation is $\sigma _ { Y } ^ { 2 } = \sum _ { i = 1 } ^ { k } y _ { i } ^ { 2 } p _ { i } - \mu _ { y } ^ { 2 } .$ Prove that the two formulas are the same.

The central limit theorem states that a. the sampling distribution of $\frac { \bar { Y } - \mu _ { Y } } { \sigma _ { \bar { Y } } }$ is approximately normal. b. $\bar { Y } \stackrel { p } { \rightarrow } \mu _ { Y }$ . c. the probability that $\bar { Y }$ is in the range $\mu _ { Y } \pm c$ becomes arbitrarily close to one as $n$ increases for any constant $c > 0$ . d. the $t$ distribution converges to the $F$ distribution for approximately $n > 30$ .

What would the correlation coefficient be if all observations for the two variables were on a curve described by $Y = X ^ { 2 } ?$

The accompanying table lists the outcomes and the cumulative probability distribution for a student renting videos during the week while on campus. $\begin{array}{l}\text { Video Rentals per Week during Semester }\\\begin{array} { | l | l | l | l | l | l | l | l | } \hline \begin{array} { l } \text { Outcome (number of weekly } \\\text { video rentals) }\end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Probability distribution } & 0.05 & 0.55 & 0.25 & 0.05 & 0.07 & 0.02 & 0.01 \\\hline\end{array}\end{array}$ Sketch the probability distribution.Next, calculate the cumulative probability distribution for the above table.What is the probability of the student renting between 2 and 4 a week? Of less than 3 a week?

To standardize a variable you

When there are $\infty$ degrees of freedom, the $t _ { \infty }$ distribution

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of a. $\operatorname { Pr } ( Y = y \mid X = x ) = \operatorname { Pr } ( Y = y )$ . b. knowing the value of one of the variables provides no information about the other. c. if the conditional distribution of $Y$ given $X$ equals the marginal distribution of $Y$ . d. $\quad E ( Y ) = E [ E ( Y \mid X ) ]$ .

A few years ago the news magazine The Economist listed some of the stranger explanations used in the past to predict presidential election outcomes.These included whether or not the hemlines of women's skirts went up or down, stock market performances, baseball World Series wins by an American League team, etc.Thinking about this problem more seriously, you decide to analyze whether or not the presidential candidate for a certain party did better if his party controlled the house.Accordingly you collect data for the last 34 presidential elections.You think of this data as comprising a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population.You generate the accompanying table: Joint Distribution of Presidential Party Affiliation and Party Control of House of Representatives, 1860-1996 $\begin{array}{|c|c|c|c|} \hline & \begin{array}{c}\text { Democratic Control } \\\text { of House }(Y=0)\end{array} & \begin{array}{c}\text { Republican Control } \\\text { of House }(Y=1)\end{array} & \text { Total } \\\hline \begin{array}{c}\text { Democratic } \\\text { President }(X=0)\end{array} & 0.412 & 0.030 & 0.441 \\\hline \begin{array}{c}\text { Republican } \\\text { President }(X=1)\end{array} & 0.176 & 0.382 & 0.559 \\\hline \text { Total } & 0.588 & 0.412 & 1.00\\\hline\end{array}$ (a)Interpret one of the joint probabilities and one of the marginal probabilities.

Show in a scatterplot what the relationship between two variables X and Y would look like if there was (a)a strong negative correlation