Exam 5: Discrete Random Variables

Obtain the probability distribution of the random variable. -When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. $(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)$ $(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)$ $(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)$ $(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)$ $(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)$ $(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)$ Let $X$ denote the smaller of the two numbers. If both dice come up the same number, then $X$ equals that common value. Find the probability distribution of $X$ . Leave your probabilities in fraction form.

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places. -The probability that a car will have a flat tire while driving through a certain tunnel is 0.00004. Use the Poisson approximation to the binomial distribution to find the probability that among 16,000 Cars passing through this tunnel, exactly two will have a flat tire.

Find the standard deviation of the binomial random variable. -The probability of winning a certain lottery is 1/70,366. For people who play 929 times, find the standard deviation for the random variable X, the number of wins.

Obtain the probability distribution of the random variable. -When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. $( 1,1 ) ( 1,2 ) ( 1,3 ) ( 1,4 ) ( 1,5 ) ( 1,6 )$ $( 2,1 ) ( 2,2 ) ( 2,3 ) ( 2,4 ) ( 2,5 ) ( 2,6 )$ $( 3,1 ) ( 3,2 ) ( 3,3 ) ( 3,4 ) ( 3,5 ) ( 3,6 )$ $( 4,1 ) ( 4,2 ) ( 4,3 ) ( 4,4 ) ( 4,5 ) ( 4,6 )$ $( 5,1 ) ( 5,2 ) ( 5,3 ) ( 5,4 ) ( 5,5 ) ( 5,6 )$ $( 6,1 ) ( 6,2 ) ( 6,3 ) ( 6,4 ) ( 6,5 ) ( 6,6 )$ Let $X$ denote the product of the two numbers. Find the probability distribution of $X$ . Leave your probabilities in fraction form.

Solve the problem. -A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a Poisson distribution with parameter $\lambda$ = 3.3. Construct a probability table for the random variable X. Compute the probabilities for 0 - 5 sightings.

Evaluate the expression. - $\left( \begin{array} { c } 12 \\ 0 \end{array} \right)$

Find the indicated probability. Round to four decimal places. -Find the probability of at least 2 girls in 6 births. Assume that male and female births are equally likely and that the births are independent events.

Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials.Let X denote the total number of successes. Round to three decimal places. - $\mathrm { n } = 4 , \mathrm { p } = \frac { 1 } { 4 } , \mathrm { P } ( \mathrm { X } = 3 )$

Find the specified probability distribution of the binomial random variable. -In one city, 25% of the population is under 25 years of age. Three people are selected at random from the city. Find the probability distribution of X, the number among the three that are under 25 Years of age.

Find the indicated binomial probability. Round to five decimal places when necessary. -A multiple choice test has 30 questions, and each has four possible answers, of which one is correct. If a student guesses on every question, find the probability of getting exactly 12 correct.

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places. -The probability that a car will have a flat tire while driving through a certain tunnel is 0.00005. Use the Poisson approximation to the binomial distribution to find the probability that among 8000 cars Passing through this tunnel, at most two will have a flat tire.

Find the mean of the random variable. -The random variable X is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate office. Its probability distribution is given in the table. Round the answer to two decimal Places when necessary. 0 1 2 3 4 5 6 7 (=) 0.24 0.01 0.12 0.16 0.01 0.14 0.11 0.21

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. -The number of power failures experienced by the Columbia Power Company in a day has a Poisson distribution with parameter $\lambda$ = 0.210. Find the probability that there are exactly two Power failures in a particular day.

Find the indicated probability. Round to four decimal places. -A machine has 7 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find The probability that the machine will be working.

Find the standard deviation of the binomial random variable. -According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size

Give an example of a discrete random variable whose possible values form a countable infinite set of numbers.

The probability distribution of a random variable is given along with its mean and standard deviation. Draw aprobability histogram for the random variable; locate the mean and show one, two, and three standard deviationintervals. -The random variable X is the number of tails when four coins are flipped. Its probability distribution is as follows. x 0 1 2 3 4 P(X=x) $\mu = 2 , \sigma = 1$

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places. -The rate of defects among CD players of a certain brand is 1.6%. Use the Poisson approximation to the binomial distribution to find the probability that among 450 such CD players received by a Store, there are exactly three defectives.

Find the standard deviation of the random variable. -The probabilities that a batch of 4 computers will contain $0,1,2,3$ , and 4 defective computers are $0.5997,0.3271,0.0669,0.0061$ , and $0.0002$ , respectively. Round the answer to two decimal places.

Find the mean of the random variable. -The random variable X is the number of golf balls ordered by customers at a pro shop. Its probability distribution is given in the table. Round the answer to two decimal places when Necessary. 3 6 9 12 15 (=) 0.14 0.29 0.36 0.11 0.10