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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Mean, Variance, and Standard Deviation In Exercises 27 and 28, find the mean, variance, and standard deviation of the discrete random variable x. See Examples 4, 5, and 6.

Spinner A spinner is divided into three equal sections labeled 1, 2. and 3. x is (a) the number landed on when the spinner is spun once and (b) the sum of the numbers landed on when the spinner is spun twice.

(a)

To determine

To calculate: The mean, variance and standard deviation of the discrete random variable x where x is the number landed on spinner when the spinner is spun once.

Explanation

Given Information:

A spinner is divided into three equal sections named 1, 2, 3.

Formula used:

Probability distribution:

The probability of a random variable x is

P(x)=Frequency of xNumber of outcomes in S=n(x)n(S)

Expected value:

If y is a discrete random variable with values y1,y2,,yn then the mean μ of the distribution is the expected value of y given by

E(y)=y1P(y1)+y2P(y2)++ynP(yn)

Where the expected value of y is denoted by E(y).

Variance:

Consider a random variable whose possible values are {x1,x2,x3,,xm} with a mean of μ. Then, the variance of the random variable is,

V(x)=(x1μ)2P(x1)+(x2μ)2P(x2)++(xmμ)2P(xm)

Standard deviation:

The standard deviation of the random variable is

σ=V(x)

Calculation:

A spinner is divided into three equal sections named 1, 2, 3.

Let S be the sample space. Then, S={1,2,3}.

So, a number of possible outcomes is 3.

The frequency of each possible outcome is,

Random variable,x123Frequency of x, n(x)111

So, the probability that 1 land on when the spinner is spun once is,

P(1)=Frequency of 1Number of outcomes in S=n(1)n(S)=13

The probability that 2 land on when the spinner is spun once is,

P(2)=Frequency of 2Number of outcomes in S=n(2)n(S)=

(b)

To determine

To calculate: The mean, variance and standard deviation of the discrete random variable x where x is the sum of the numbers landed on spinner when the spinner is spun twice.

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