The probability distribution of the random variable X represents the number of h

Question

The probability distribution of the random variable X represents the number of hits a baseball player obtained in a game for the 2012 baseball season.

x          P(x)

0          0.1668

1          0.3446

2          0.2907

3          0.1496

4          0.0372

5          0.0111

The probability distribution was used along with statistical software to simulate 25 repetitions of the experiment (25 games). The number of hits was recorded. Approximate the mean and standard deviation of the random variable X based on the simulation. The simulation was repeated by performing 50 repetitions of the experiment. Approximate the mean and standard deviation of the random variable. Compare your results to the theoretical mean and standard deviation. What property is being illustrated?

A. Compute the theoretical mean of the random variable x for the given probability distribution

B. Compute the theoretical standard deviation of the random variable x for the given probability distribution

C. Approximate the mean of the random variable x based on the simulation for 25 games

D. Approximate the standard deviation of the random variable x based on the simulation for 50 games.

E. Approximate the standard deviation of the random variable x based on the simulation for 50 years

F. What property is being illustrated

Explanation / Answer

Solution:

a.Compute the theoretical mean of the random variable X for the given probability distribution.

Theoretical mean, mu = sum x_{i}p(x_{i})

= 0*0.1668 + 1*0.3446 + 2*0.2907 + 3*0.1496 + 4*0.0372 + 5*0.0111

= 1.5791

b. Compute the theoretical standard deviation of the random variable X for the given probability distribution.

Theoretical variance, Var(x) = sum (x_{i}-mu)^{2}p(x_{i})

=(0-1.5791)^{2}*0.1668 + (1-1.5791)^{2}*0.3446 + (2-1.5791)^{2}*0.2907 + (3-1.5791)^{2}*0.1496 + (4-1.5791)^{2}*0.0372 + (5-1.5791)^{2}*0.0111

= 0.982407

Theoretical Standard deviation, sigma = sqrt{ 0.982407} = 0.991164

c.Approximate the mean of the random variable X based on the simulation for 25 games.

Used statistical software R to get the below values of X for 25 games based on the probability distribution p(x).

X = 1 2 4 1 0 4 1 4 0 0 2 3 2 2 2 0 3 5 0 3 2 0 2 2 1

Xbar = (1 +2 +4 +1 +0 +4 +1 +4 +0 +0 +2 +3 +2 +2 +2 +0 +3+ 5+ 0 +3+ 2+ 0 +2 +2 +1)/25

= 1.84

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