Question

6. Discrete probability distributions #2

You may calculate the answers of the following questions using the equations in your textbook for the appropriate probability distribution function. (You will need to read the questions first to determine the appropriate distribution.) However, you may also determine these answers by using the Distributions tool. Select the appropriate distribution from the dropdown box in the upper left-hand corner. Set the parameters accordingly, and select the radio button in the lower left-hand corner depending on whether you want the probability of a single outcome (left) or a cumulative probability (right).

The Geminids is an annual meteor shower that appears every December. Under a clear, dark sky, an observer of the Geminids would see an average of 20 meteors per 10-minute period (if the meteors’ emanation point were directly overhead).

a. It’s December and you host a Geminids party on the peak night of the meteor shower. The sky is clear and dark, and the meteors’ emanation point is directly overhead. You and your friends watch the sky for 10 minutes. The probability that you see exactly 20 meteors is______.

0.4409

0.1322

0.0309

0.0888

b. The probability that you see more than 16 meteors while watching the night sky for 10 minutes is_________.

0.0646

0.7789

0.4634

0.8435

c. After going inside for a midnight snack, you and your friends go back outdoors for a 5-minute sky-gazing session. The probability that you observe no more than 14 meteors during this sky-gazing session is ______.

0.7973

0.6618

0.9165

0.0835

d. The number of meteor sightings over 20 minutes has an expected value of________ and a standard deviation of_______.

-----20,14,7.6,40

-----6.32,2.76,40,3.74

Answer 1

a)

an observer of the Geminids would see an average of 20 meteors per 10-minute period

so,

Mean/Expected number of events of interest: λ =       20 / minute

poisson probability distribution

P(X=x) = e-λλx/x!

p(eexactly 20 ) = P ( X =    20 )

= (e ^(-20) *    20²⁰ ) /   20 !

=0.0888

The probability that you see exactly 20 meteors is______0.888

.............

b)

p(more than 16) = 1- p(x<=16)

= 1-  e ^-20 * 20 ¹⁶ /   16 !

= 1-  0.0646

= 0.9354

The probability that you see more than 16 meteors while watching the night sky for 10 minutes is_ = 0.9354

.................

c)

average number of meteors per 5 minute period = 10meteors per 5 minutes

λ = 10

p(no more than 14) = p(x <=14)

= 0.9165

The probability that you observe no more than 14 meteors is = 0.9165

.............

d)

for 20 minutes sightings ,     λ = 40 meteros / 20minutes

in poisson distribution , mean = λ

so,

The number of meteor sightings over 20 minutes has an expected value = 40

variance = λ = 40

std dev = √λ =    6.32

please revert back for doubt