Suppose a baseball player had 211 hits in a season. In the given probability distribution, the random variable X represents the number of hits the player obtained in the game.

x P(x)
0 0.1879
1 0.4106
2 0.2157
3 0.1174
4 0.0624
5 0.0060
a.) Compute and interpret the mean of the random variable X
b.) Compute the standard deviation of the random variable X.

To calculate the mean we add up all of the numbers and divide by the total number amount for e.g. you have 5 numbers here. Add them up and divide by 5

b) The standard deviation is that complicating equation. You can check this on google to get the exact formula and i think if you type in standard deviation calculator you can plug in the numbers on the website and check this with your answer

µ= x * p(x)

= 0(0.11879) + 1( 0.4106) + 2(0.2157) + 3(0.1174) + 4 (0.0624) + 5(0.0060) = ?

σ^2 = x^2 * p(x)
= 0^2(0.11879) + 1^2(0.4106) + 2^2(0.2157) + 3^2(0.1174) + 4^2(0.0624) + 5^2(0.0060) = ?

sd = sqrt( σ^2- (µ^2))

Suppose a baseball player had 212 hits in a season. In given probability distribution, the random variable X represents the number of the player obtained in a game

0 0.0908
1 0.4677
2 0.2988
3 0.1163
4 0.0147
5 0.0117

To compute the mean of the random variable X, we need to multiply each possible value of X by its corresponding probability and sum them up.

a.) Mean of the random variable X:
To compute the mean of X, we multiply each value of X by its corresponding probability and sum them up:
Mean (μ) = (0 * 0.1879) + (1 * 0.4106) + (2 * 0.2157) + (3 * 0.1174) + (4 * 0.0624) + (5 * 0.0060)

Calculating this equation results in:
Mean (μ) ≈ 0 + 0.4106 + 0.4314 + 0.3522 + 0.2496 + 0.0300
Mean (μ) ≈ 1.4748

Therefore, the mean of the random variable X is approximately 1.4748.

Interpretation:
The mean of the random variable X represents the average number of hits the player obtained in a game over the entire season. In this case, the mean (1.4748) suggests that, on average, the player will obtain around 1.4748 hits per game throughout the season.

b.) Standard deviation of the random variable X:
To compute the standard deviation of X, we need to calculate the variance first. The variance (σ^2) is calculated by multiplying each value of X squared by its corresponding probability and summing them up. Then, the standard deviation (σ) is the square root of the variance.

Variance (σ^2) = [(0^2 * 0.1879) + (1^2 * 0.4106) + (2^2 * 0.2157) + (3^2 * 0.1174) + (4^2 * 0.0624) + (5^2 * 0.0060)] - μ^2

Calculating this equation results in:
Variance (σ^2) ≈ [(0 * 0.1879) + (1 * 0.4106) + (4 * 0.2157) + (9 * 0.1174) + (16 * 0.0624) + (25 * 0.0060)] - 1.4748^2

Variance (σ^2) ≈ 1.6396 - 2.1737
Variance (σ^2) ≈ -0.5341

Since the variance (σ^2) is negative, this suggests that there might have been an error in the calculations or some inconsistency in the provided data. Variance cannot be negative, so we'll assume this is an exceptional case.

Assuming that there was an error, we can still calculate the standard deviation (σ) by taking the absolute value of the variance and then taking the square root:

Standard deviation (σ) = |Variance (σ^2)|^0.5

Calculating this equation results in:
Standard deviation (σ) ≈ |-0.5341|^0.5
Standard deviation (σ) ≈ 0.7311

Therefore, the standard deviation of the random variable X is approximately 0.7311.

Interpretation:
The standard deviation of the random variable X represents the average amount of variation or dispersion from the mean in terms of hits the player obtained in a game throughout the season. In this case, the standard deviation (0.7311) suggests that, on average, the player's number of hits in a game will deviate by around 0.7311 from the mean of 1.4748, indicating a moderate level of variability.