Question

(i) A binomial random variable has a mean equal to 187 and a standard deviation of 12.

(a) Find the value of n. (Give your answer correct to the nearest whole number.)

(b) Find the value of p. (Give your answer correct to four decimal places.)

(ii) he probability of success on a single trial of a binomial
experiment is known to be 0.3. The random variable *x*,
number of successes, has a mean of 75.

(a) Find the number of trials involved in this experiment. (Give
your answer correct to the nearest whole number.)

(b) Find the standard deviation of *x*. (Give your answer
correct to two decimal places.)

Answer #1

i)

For binomial distribution ,

Given, Mean = 187 , Standard deviation = 12

That is

np = 187 and Sqrt( np(1-p) ) = 12

Put the value of np from first equation in second and solve for p

Sqrt( np (1-p) ) = 12

Sqrt ( 187 ( 1 - p) ) = 12

Take square of both sides

187 ( 1 - p) = 144

1 - p = 144 / 187

p = 1 - 144/187

**p = 0.2299**

So,

np = 187

n * 0.2299= 187

**n = 813**

a)

n = 813

b)

p = 0.2299

ii)

a)

Given, p = 0.3 and mean = np = 75

So,

n * 0.3 = 75

n = 75 / 0.3

n = **250**

b)

Standard deviation = sqrt( np(1-p) )

= sqrt( 250 * 0.3 * 0.7)

= **7.25**

Assume the random variable X has a binomial distribution with
the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
P(X>4), n=7, p=0.4
Find the standard deviation of the following data. Round your
answer to one decimal place.
x
1
2
3
4
5
6
P(X=x)
0.1
0.1
0.2
0.1
0.2
0.3

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the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
P(X<2), n=5 p=0.3

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the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
P(X < 4) , n = 8, p = 0.3

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the given probability of obtaining a success. Find the following
probability given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
P(X=4), n=6, p=0.3

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the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
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the given probability of obtaining success. Find the following
probability, given the number of trials and the probability of
obtaining success. Round your answer to four decimal places.
P(X≥7), n=10, p=0.3

A random variable has a 0.02
probability of success in each independent trial, where the total
number of trials is n= 90.
a. What
is the expected number of successes in 90 trials?
b. What
is the standard deviation of successes in 90 trials?
c. Use
the binomial distribution to find the probability of 90 trials.
d. Use
the Poisson distribution to approximation find the probability of
in
90
trials.

Assume the random variable X has a binomial distribution with
the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.

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the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
obtaining a success. Round your answer to four decimal places.
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the given probability of obtaining a success. Find the following
probability, given the number of trials and the probability of
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