1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Schach [20]
2 years ago
11

Let the number of chocolate chips in a certain type of cookie have a Poisson distribution. We want the probability that a cookie

of this type contains at least two chocolate chips to be greater than 0.99. Find the smallest value of the mean that the distribution can take.

Mathematics
1 answer:
ludmilkaskok [199]2 years ago
4 0

Answer:

\lambda \geq 6.63835

Step-by-step explanation:

The Poisson Distribution is "a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event".

Let X the random variable that represent the number of chocolate chips in a certain type of cookie. We know that X \sim Poisson(\lambda)

The probability mass function for the random variable is given by:

f(x)=\frac{e^{-\lambda} \lambda^x}{x!} , x=0,1,2,3,4,...

And f(x)=0 for other case.

For this distribution the expected value is the same parameter \lambda

E(X)=\mu =\lambda

On this case we are interested on the probability of having at least two chocolate chips, and using the complement rule we have this:

P(X\geq 2)=1-P(X

Using the pmf we can find the individual probabilities like this:

P(X=0)=\frac{e^{-\lambda} \lambda^0}{0!}=e^{-\lambda}

P(X=1)=\frac{e^{-\lambda} \lambda^1}{1!}=\lambda e^{-\lambda}

And replacing we have this:

P(X\geq 2)=1-[P(X=0)+P(X=1)]=1-[e^{-\lambda} +\lambda e^{-\lambda}[]

P(X\geq 2)=1-e^{-\lambda}(1+\lambda)

And we want this probability that at least of 99%, so we can set upt the following inequality:

P(X\geq 2)=1-e^{-\lambda}(1+\lambda)\geq 0.99

And now we can solve for \lambda

0.01 \geq e^{-\lambda}(1+\lambda)

Applying natural log on both sides we have:

ln(0.01) \geq ln(e^{-\lambda}+ln(1+\lambda)

ln(0.01) \geq -\lambda+ln(1+\lambda)

\lambda-ln(1+\lambda)+ln(0.01) \geq 0

Thats a no linear equation but if we use a numerical method like the Newthon raphson Method or the Jacobi method we find a good point of estimate for the solution.

Using the Newthon Raphson method, we apply this formula:

x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)}

Where :

f(x_n)=\lambda -ln(1+\lambda)+ln(0.01)

f'(x_n)=1-\frac{1}{1+\lambda}

Iterating as shown on the figure attached we find a final solution given by:

\lambda \geq 6.63835

You might be interested in
Terrell is planning a trip he begins looking for plane tickets on Monday the cost is $415.65 the website alerts fin that prices
just olya [345]

Answer:

536.74

Step-by-step explanation:

I think so

7 0
3 years ago
The following data set represents the number of siblings of six students in music class
Julli [10]

Answer:

I tried my best to made the graph

3/-1/2

2/0/1

4/-2/3

2/0/1

1/+1/0

0/+2/1

4 0
3 years ago
The value of y is directly proportional to the value of x. If y =35 when x = 140, what is the value of y when x =70?
torisob [31]
17.5 because 140 divided by 35 is 17.5 and so is 70 divided by 35.
3 0
2 years ago
Find the range of values of p if the median is 3
faltersainse [42]

If we deal with Natural numbers then the range is

1 ≤ p ≤ 5

4 0
3 years ago
Help please I need it
pashok25 [27]

Answer:

come here I help see my questions

4 0
3 years ago
Other questions:
  • Is x=-2 calculate the value of x2 +2x
    14·1 answer
  • Coats are regularly priced at $34.00. They are now on sale for $14.28. If Taylor buys 4, then how much will Taylor spend?
    15·1 answer
  • Which function below has the lowest y intercept? f(x) graph going through (0, 5) and (2, 1) g(x) Kyle started the summer having
    15·2 answers
  • Pwlc is definitely a parallelogram
    10·1 answer
  • What is the sum of the fractions below?<br> 1/7 + 3/8
    5·1 answer
  • a photograph has a perimeter of 24 in the difference between the photographs Lane and the width is 2 inches. find the length and
    15·1 answer
  • What is the population density?​
    11·1 answer
  • Complete the identity sin(2θ) sin(4θ) cos(2θ) cos(4θ) = ?
    10·1 answer
  • 1)<br> Simplify: -7(2m - 8) + 6m<br> 2)<br> Expand: -10(1 - 6r)
    6·1 answer
  • An ice cream store sells 27 flavors of ice cream. Determine the number of 6 dip sundaes. How many 6 dip sundaes are possible if
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!