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
Anika [276]
3 years ago
12

Find the mean, variance &a standard deviation of the binomial distribution with the given values of n and p.

Mathematics
1 answer:
MrMuchimi3 years ago
7 0
A random variable following a binomial distribution over n trials with success probability p has PMF

f_X(x)=\dbinom nxp^x(1-p)^{n-x}

Because it's a proper probability distribution, you know that the sum of all the probabilities over the distribution's support must be 1, i.e.

\displaystyle\sum_xf_X(x)=\sum_{x=0}^n\binom nxp^x(1-p)^{n-x}=1

The mean is given by the expected value of the distribution,

\mathbb E(X)=\displaystyle\sum_xf_X(x)=\sum_{x=0}^nx\binom nxp^x(1-p)^{n-x}
\mathbb E(X)=\displaystyle\sum_{x=1}^nx\frac{n!}{x!(n-x)!}p^x(1-p)^{n-x}
\mathbb E(X)=\displaystyle\sum_{x=1}^n\frac{n!}{(x-1)!(n-x)!}p^x(1-p)^{n-x}
\mathbb E(X)=\displaystyle np\sum_{x=1}^n\frac{(n-1)!}{(x-1)!((n-1)-(x-1))!}p^{x-1}(1-p)^{(n-1)-(x-1)}
\mathbb E(X)=\displaystyle np\sum_{x=0}^n\frac{(n-1)!}{x!((n-1)-x)!}p^x(1-p)^{(n-1)-x}
\mathbb E(X)=\displaystyle np\sum_{x=0}^n\binom{n-1}xp^x(1-p)^{(n-1)-x}
\mathbb E(X)=\displaystyle np\sum_{x=0}^{n-1}\binom{n-1}xp^x(1-p)^{(n-1)-x}

The remaining sum has a summand which is the PMF of yet another binomial distribution with n-1 trials and the same success probability, so the sum is 1 and you're left with

\mathbb E(x)=np=126\times0.27=34.02

You can similarly derive the variance by computing \mathbb V(X)=\mathbb E(X^2)-\mathbb E(X)^2, but I'll leave that as an exercise for you. You would find that \mathbb V(X)=np(1-p), so the variance here would be

\mathbb V(X)=125\times0.27\times0.73=24.8346

The standard deviation is just the square root of the variance, which is

\sqrt{\mathbb V(X)}=\sqrt{24.3846}\approx4.9834
You might be interested in
The sum of a rational number and an irrational number equals:
WINSTONCH [101]

Answer: is an irrational number

Step-by-step explanation:

Like adding three to pi (3.14159265358979323846264....)is still going to be irrational

3 0
3 years ago
Read 2 more answers
A circle has a radius of 11 inches and a central angle AOB that measures 45°. What is the area of sector AOB? Use 3.14 for pi an
Vesna [10]
The answer is A from what my teacher taught me
7 0
3 years ago
Read 2 more answers
The larger leg of a right triangle is 3cm longer than its smaller leg. The hypotenuse is 6cm longer than the smaller leg. how ma
Natalka [10]
b=a+3\\
c=a+6\\
a^2+b^2=c^2\\\\
a^2+(a+3)^2=(a+6)^2\\
a^2+a^2+6a+9=a^2+12a+36\\
a^2-6a-27=0\\
a^2-9a+3a-27=0\\
a(a-9)+3(a-9)=0\\
(a+3)(a-9)=0\\
a=-3 \vee a=9\\


The length can't be negative, so the answer is 9 cm.

5 0
3 years ago
Read 2 more answers
What is 13 hours after 4:00 am.
kupik [55]
I believe it would be 5:00 pm, because 12 hours after 4:00 am is 4:00 pm, plus one hour is 5:00pm.
8 0
3 years ago
Read 2 more answers
Find the sale price of a $45 item after a 20% discount.
devlian [24]

Answer:

36$

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • What does least common multiple
    7·1 answer
  • In the number 555,55 each digit is a 5 but value of rachis different based on its place in the number. How does the value of 5 i
    13·1 answer
  • Six minus a number divided by 3 equals 8. What is the number?
    14·2 answers
  • Tom begins his morning shift at 9:45
    12·1 answer
  • Cheryl wants to find the number of hours seventh-graders do homework each week. There are 297 girls and boys in seventh grade. W
    5·1 answer
  • What is the midpoint of the line segment With end points 0,4 and -4,-6?
    11·1 answer
  • HELP ME !
    11·1 answer
  • Need help please and thank you
    14·1 answer
  • (x-2)= - 1/4(x-8) what does x equal?
    9·2 answers
  • Mrs. Spring left half of her estate to her son Sam. She left half of the remaining half to her cousin Fred. She left half of the
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!