Exam 12: Probability and Calculus

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) = F(b)

The probability density function for a random variable X is f(x) = $\frac { 3 } { 4 }$ (2x - $x ^ { 2 }$ ), $0 \leq x \leq 2$ . Find $\operatorname { Pr } ( 0 \leq X \leq 1 )$ Enter just a reduced fraction.

When mice are placed in a certain maze the amount of time it takes them to go through the maze is approximately normally distributed with a mean of 25 minutes and a standard deviation of 5 minutes. What is the probability that a mouse will complete the maze in under 30 minutes? (Hint: find the normal density function first). Enter just a real number rounded off to two decimal places.

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

A person throws a die until the side with two spots appears. The probability of throwing the die exactly n times before throwing a "2" is $\left( \frac { 5 } { 6 } \right) ^ { n }$ $\left( \frac { 1 } { 6 } \right)$ , n ≥ 0. What is the probability that the number of throws before throwing a "2" is even? Enter just a reduced fraction.

A random variable X is exponentially distributed with a mean of 2. Find $\operatorname { Pr } ( 1 \leq X \leq 3 )$

A car dealer records the number of Mercedes sold each week. During the past 50 weeks, there were 15 weeks with no sales, 20 weeks with one sale, 10 weeks with two sales, and 5 weeks with three sales. Let X be the number of Mercedes sold in a week selected at random from the past 50 weeks. Compute E(X). Enter just a real number rounded off to one decimal place (no label).

Determine the probability of an outcome of the probability density function $f ( x ) = 12 x ^ { 2 } - 12 x ^ { 3 }$ being between $\frac { 1 } { 2 } \text { and } 1$ where $0 \leq x \leq 1$ . Enter just a reduced fraction.

Joe has a lawn mowing job. If he completes the work he earns $40. But there is a 30% chance it may rain, in which case he won't finish the job. He can pay Jane $20 to help him and ensure that he finishes the job. If X is the amount Joe will get if he does not get Jane to help, calculate E(X) and thus decide whether Joe should hire Jane or not. (If it rains, assume Joe will make no money and if Joe hires Jane assume they will be able to finish the job before it rains. Enter your answer exactly as a,b where a is an integer representing E(X) in dollars (no units), and b is either "yes" or "no" answering the question "should Joe hire Jane?", separated by a comma.

A lumber yard cuts 2" x 4" lumber into 8 foot studs. It is observed that the actual lengths of the studs are normally distributed with mean 8 feet and standard deviation 1 foot. What proportion of the studs are longer than 8.25 feet? Enter just a real number rounded off to two decimal places.

The table below is the probability table for a random variable X. Find the standard deviation of X. Outcome -1 0 1 2 Probability Enter just a real number rounded off to two decimal places.

Let X be the time to failure of an electronic component, and suppose X is an exponential random variable with $E ( X ) = 4 \text { years }$ . Find the median lifetime, i.e., find M such that $\operatorname { Pr } ( X \leq M ) = \frac { 1 } { 2 }$ . Enter just a real number rounded to two decimal places (no units).

A random variable X has a probability density function f(x) = $\frac { x } { 32 }$ , $0 \leq x \leq 8$ . Find a such that $\operatorname { Pr } ( X \geq a ) = \frac { 1 } { 4 }$ Enter your answer exactly in the reduced form b $\sqrt { c }$ , unlabeled.

A Christmas tree grower anticipates a profit of $80,000 in a usual season. There is however a 10% chance of pine bark beetle infestation in which case 70% of the trees are destroyed and profit is reduced to $24,000. The grower can spray for beetles at the beginning of the season at a cost of $7,000. Compute E(X). Enter just an integer rounded off to the nearest thousand.

Suppose f(x) = k( $x ^ { 2 }$ + 2x) is a probability density function for a continuous random variable on the interval $0 \leq x \leq 3$ Find the value of k and find the corresponding cumulative distribution function. Enter just an unlabeled polynomial in x in standard form.

A random variable has probability density function f(x) = 30 $x ^ { 2 }$ (1 - x $)^2$ (0 ≤ x ≤ 1). Compute its cumulative distribution F(x).

Given the probability density function f(x) = $\frac { 1 } { 3 }$ , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15. Enter an unlabeled polynomial in x in standard form.

If f(x) = $\frac { 1 } { 8 }$ x is a probability density function for 0 ≤ x ≤ 4, find F(x), the corresponding cumulative distribution function and use it to find Pr(1 ≤ X ≤ 3). Enter just a reduced fraction representing $\operatorname { Pr } ( 1 \leq X \leq 3 )$ Do not label.

The probability density function for a random variable X is $f ( x ) = \frac { 2 \ln x } { ( \ln 4 ) ^ { 2 } x }$ , $1 \leq x \leq 4$ Find $\operatorname { Pr } ( 1 \leq X \leq 2 )$ Enter just a reduced fraction.

Suppose the number of cars passing through a toll booth in a 10 minute interval is a Poisson random variable. If the average number of cars is 23, give an expression for the probability that n cars pass through the booth. Is $p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }$ correct?