Project #17008 - Statistics & Applications

CEE3770 Statistics & Applications

Problems ON Probability, geometric probability, conditional probability, BINOMIAL,

POISSON, EXPONENTIAL and GAUSSIAN DISTRIBUTIONS

1) The continuous random variable X is defined over the interval [-1,1]. Consider the function

f (x) = ax2 +1/ 5.

Determine a such that f(x) is the probability density function of X. Find mean and variance of X.

____________________________________________________________________

2) A fair coin is flipped 5 times. What is the probability of getting 3 heads? What is the

probability of getting 5 tails? If the coin is tossed 100 times, what is the expected average of

heads ?

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4) Consider the discrete random variable X=’ the product of the outcomes from rolling two dice

at the same time’. Assume a fair dice with 4 faces. Determine:

a. The sample space S associated with the random variable X;

b. The probability function of X, that is p(x)=Pr[X =x];

c. the mean and standard deviation of X;

d. the probability that X >6.

______________________________________________________________________

5) Pick a point x at random (with uniform density) in the interval [-4 4].

a) Find the probability that x < 1/3.

b) Find the probability that x2- 4<0.

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6) At the Atlanta airport during 1 day, John observed 500 airplane arrivals. He noted that in

25 occasions the waiting time between 2 successive arrivals was more than 15 minutes.

What was the average waiting time between arrivals that day at the airport ?

______________________________________________________________________

7) Erica tosses a coin 1000 times for an experiment in her statistics course. She found that tail

occurred only 50 times during the experiment. Is her coin a ‘fair coin’? Provide a mathematical

argument based on probabilistic principles.

8) Chris tosses a coin 1000 times. Assume that he flips a fair coin. What is the probability that

more than 908 heads occur?

____________________________________________________________________

9) Pick a point x at random (with uniform density) in the interval [0 1]. Find the probability that x

> 1/2, given that x > 1/3.

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10) Pick a point x at random (with uniform density) in the interval [0 1]. Find the probability that

x2- 1/2>0.

______________________________________________________________________

11) Suppose you toss a dart at a circular target of radius 20 cm. Given that the dart lands in the

upper half of the target, find the probability that

a. it lands in the right half of the target.

b. its distance from the center is less than 5 cm.

c. its distance from the center is greater than 5 cm.

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12) A fair coin is tossed 3 times in a row. Use Kolmogorov theory and identify the sample

space S using the 23 arrangements for 3 tosses. i) Clearly identify all the sets of S and

assign a probability assuming equiprobable events. ii) Then compute the probability

associated to the following events (sets) (do not use combinatorics)

{0 3 } 0 A = heads after tosses

{1 3 } 1 A = head after tosses

{2 3 } 2 A = headsafter tosses

{3 3 } 3 A = heads after tosses .

iii) Then, show that

0 1 2 3 S = A ∪ A ∪ A ∪ A

and that the partition events Aj are mutually exclusive, i.e. they are disjoint sets.

iv) Then compute the probabilities P(Aj) using just combinatorics.

______________________________________________________________________

13) A fair coin is tossed 10 times in a row. What is the probability of getting at least 2 heads ?

(Hint: use Binomial distribution)

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14) Consider L as the length of a pen manufactured by a company. L is Gaussian with

mean=20 cm and standard deviation=0.02 cm. If the company produces 1,000,000 pens

in 1 month, use tables to answer the following questions:

a) How many pens will have an height greater than 20.0 cm ?

b) How many pens will have an height smaller than 20.04 cm ?

______________________________________________________________________

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15) A fair coin is tossed 3 times in a row. Consider the ramdom variable X=’number of heads

after 3 tosses’.

a. Define the sample space S and the associated probability function (pf). Plot pf.

b. Plot the associated cumulative distribution function F(x)=Pr[X<=x] (cdf);

c. Compute mean and standard deviation of X.

Hint: The sample space S={ ‘0 heads in 3 tosses’, ‘1 head in 3 tosses’ , ‘2 head in 3

tosses’, ‘3 heads in 3 tosses’}. Use combinatorics to compute Pr[X=0]=p(x=0),

Pr[X=1]=p(x=1), Pr[X=2]=p(x=2), Pr[X=3]=p(x=3). Verify that Pr[S]=1, that is

p(0)+p(1)+p(2) +p(3)=1.

_____________________________________________________________

16) Consider an exponential distribution to model the inter-time T between successive

arrivals at the bus station. The parameter λ=10 arrivals/hour.

a) What is the probability to wait between 5 and more than 10 minutes at the station for

the next bus ?

b) In 100 arrivals, I waited for the next bus more than 5 minutes in 10 occasions. What

is the average number of arrivals ?

Subject Mathematics
Due By (Pacific Time) 11/21/2013 12:00 am
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