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Cast a fair die and let X = 0 if 1, 2, or 3 spots appear, let X = 1 if 4 or 5 spots appear, and let X = 2 if 6 spots appear. Do this two independent times, obtaining X1 and X2. Calculate P(|X1 – X2| = 0).
Let f(x1,x2,x3)=exp[−(x1 + x2 + x3)], 0 < x1 < ∞, 0 < x2 < ∞, 0 < x3 < ∞, = zero elsewhere, be the joint pdf of X1, X2, X3. (a) Compute P (X1 < 2X2) and P (X1 = 2X2). (b) Determine the mgf of X1+2X2+X3. Are X1, X2, and X3 random variables independent? Give reasoning.
Let X and Y have the joint pdf f(x, y) = 3x, 0 < y < x < 1, zero elsewhere. Are X and Y independent? If not, find Var(Y|x).
Sam has 6 rose bushes. He counted the flowers on each of them. There are 8, 2, 5, 4, 11 and 9. Find the Standard Deviation. Is X a “Usual” number of flowers? X is your last two digits of your CUNYFirst ID Number. (00 = 0, 01 = 1, …) (20 points) my id is 03
The life of a 60- watt light bulb in hours is known to be normally distributed with σ = 25 hours. Create 5 different random samples of 100 bulbs each which has a mean life of x_bar ~ 1000 hours and perform one-way ANOVA with state it.
3. A computing system manager states that the rate of interruptions to the internet
service is 0.2 per week. Use the Poisson distribution to find the probability of
a) one interruption in 3 weeks
b) at least two interruptions in 5 weeks
c) at most one interruption in 15 weeks
4. A special cable has an average breaking strength of 800 pounds. The standard deviation of
the population is 12 pounds. A researcher selects a sample of 35 cables and founds that the
average breaking strength is 805 pounds. Assume that the variable is normally distributed.
A. Find the point estimate of the cable average breaking strength (μ)
B. Construct the 95% confidence interval for the population average breaking strength of
the cable (μ) and interpret your result.
C. Can the researcher conclude that the average breaking strength of the cable greater than
800 pounds at 1% level of significance?
1. Suppose that the algorithm, or robot reporter, typically writes proportion 0.65
of the stories on the site. If 15 new stories are scheduled to appear on a web site next
weekend, find the probability that
a) 11 will be written by the algorithm.
b) at least 10 will be written by the algorithm
c) between 8 and 11 inclusive will be written by the algorithm.
2. The amount of time it takes to assemble a computer is normally distributed, with a mean of
50 minutes and a standard deviation of 10 minutes.
a. What is the probability that a computer is assembled in a time between 45 and 60 minutes?
b. What is the probability that a computer is assembled in time more than 65 minutes?
c. What is the probability that their average time will be less than 45 minutes?
A cereal manufacturer is aware that the weight of the product in the box varies slightly from box to box. In fact, considerable historical data has allowed the determination of the density function that describes the probability structure for the weight (in ounces). In fact, letting X be the random variable weight, in ounces, the density function can be described as f(x) ={2/5, 23.75 ≤ x ≤ 26.25,
{0, elsewhere
(a) Verify that this is a valid density function.
(b) Determine the probability that the weight is smaller than 24 ounces.
(c) The company desires that the weight exceeding 26 ounces is an extremely rare occurrence.
they consider standard care. One hundred patients with hypertension are enrolled and randomized
to one of the two comparison treatments. After taking the assigned medication for 6 weeks, their
systolic blood pressure (SBP) is measured. Summary statistics are given here. Use the data to test
if there is a significant difference in systolic blood pressures between medication groups assuming
equality of the variances. Run the appropriate test at a 5% level of significance.
Treatment Number of Patients Mean SBP Standard Deviation
New 50 130 12
Standard 50 135 10
checked. The selling price of each house is noted. The following information is obtained:
Town A Town B
