Exam 12: Probability and Calculus

The table below is the probability table for a random variable X. Find the standard deviation of X. Outcome -3 -2 -1 1 2 3 Probability 0.1 0.1 0.4 0.3 0.05 0.05 Enter just a real number rounded off to two decimal places.

Which of the graphs below could not possibly be the graph of a probability function f(x)?

Let X be a continuous random variable A ? X ? B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false. - $\int _ { A } ^ { B } F ( x ) = 1$

Given the density function f(x) = $\frac { 3 } { 64 }$ $x ^ { 2 }$ , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

A survey shows that the time spent in a checkout line in a certain supermarket has an exponential density function with mean 5 minutes. What is the probability of spending 10 minutes or more in a checkout line? Enter just a real number rounded off to two decimal places.

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: $f ( x ) = \frac { 7 } { \sqrt { 2 \pi } } e ^ { - 49 / 2 ( x - 3.9 ) ^ { 2 } }$ Enter your answer as just two numbers (a real number to one decimal place followed by a reduced fraction) separated by a comma, the first representing E(X) and the second representing $\sqrt { \operatorname { Var } ( X ) }$ .

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false. -Pr(a ≤ X ≤ b) = $\int _ { a } ^ { b } f ( x ) d x$

A farmer has observed that the time to maturation of a certain crop is approximately normally distributed with a mean of 60 days and a standard deviation of 2 days. Find the percentage of plants that will mature in less than 55 days. Enter the percentage as just a real number rounded off to two decimal places followed by %.

A set of exam scores is 80, 75, 85, 90, 100, 70, 60. The standard deviation equals

Find the value of k that makes f(x) = $k x ^ { 3 }$ a probability density function on the interval $0 \leq x \leq 1$ . Enter just an integer.

Find the value of k that makes f(x) = $k x ^ { 2 }$ a probability density function on the interval $0 \leq x \leq 1 .$ Enter just an integer.

Find the expected value and variance for the random variable whose probability density function is $f ( x ) = x$ $0 \leq x \leq 1 .$ Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

The probability density function of a continuous random variable X is $f ( x ) = \frac { 3 } { 2 } x - \frac { 3 } { 4 } x ^ { 2 }$ , $0 \leq x \leq 2$ . Is this the graph of f(x) with the shaded area corresponding to $\operatorname { Pr } \left( \frac { 1 } { 2 } \leq x \leq \frac { 3 } { 2 } \right)$ ?

Suppose f(x) = $\frac { 1 } { x ^ { 2 } }$ is a probability density function for x ≥ 1. Find Pr(2 ≤ X ≤ 10). Enter just a reduced fraction.

John would like to place a two dollar bet on his favorite racehorse, Black Velvet. He can bet that Black Velvet will win or show (finish in the top three horses). If he bets correctly that Black Velvet wins, he wins $20. If he bets correctly that Black Velvet shows, he wins $7. John figures Black Velvet has a 20% chance of winning and a 70% chance of showing. If X is the amount of money John wins if he bets Black Velvet will win and Y is the amount of money he wins if Black Velvet will show, find E(X) and E(Y) . Enter just two real numbers rounded off to two decimal places in the order given above representing dollars (no units).

A table saw cuts construction studding. Observation has shown that the lengths of the studs are normally distributed with a mean of 10 feet and a standard deviation of 6 inches. Which of the following correctly represents the probability that a randomly chosen stud exceeds 11 feet?

The table below is the probability table for a random variable X. Find Var(X). Outcome -3 -2 -1 1 2 3 Probability 0.1 0.1 0.4 0.3 0.05 0.05 Enter just a real number rounded off to two decimal places.

A random variable X has a density function f(x) = $\frac { 24 } { x ^ { 3 } }$ , 3 ≤ x ≤ 6. Find b such that $\operatorname { Pr } ( X \leq b ) = 0.4$ . Enter your answer exactly in the reduced form a $\sqrt { \frac { b } { c } }$ , unlabeled.

A basketball player attempts successive free throws until he succeeds in making a basket. Suppose the probability of success of each attempt is 0.7; thus, the probability of exactly n failures before the first success is $( 0.3 ) ^ { n }$ (0.7), n ≥ 0. What is the probability that the number of failures before the first successful free throw is odd? Enter just a reduced fraction.

The riders of the New Town Elementary school bus consists of 5 five year olds, 3 six year olds, 10 eight year olds, 1 nine year old, 4 eleven year olds and a twelve year old. A child is selected at random and her age is noted. Let X be the outcome. Find E(X). Enter just a reduced fraction of form $\frac { a } { b }$ (no label).