1. mean = average
2. variance means :
if everyone got a high or low score on a test then the variance is low or 0
variance means are there a lot of different test scores ?
if there are then the variance is high
variance = 1. each different number minus the average squared 2. add them all up 3. divide the total by the total number of numbers
3. square root of variance = standard deviation
standard deviation is like the distance between a person's house and their friend's house to see how far they typically have to travel.
answers :
mean= 4.64 , variance= -2.4596 , standard deviation = 1.57
steps
rewrite values
X, P(X)
1 , 0.05
2 , 0.12
3 , 0.21
4 , 0.33
5 , 0.48
Certainly! Here are the calculations for the mean, variance, and standard deviation of the payment made under this policy:
Mean (Expected value):
E(X) = (1 * 0.05) + (2 * 0.12) + (3 * 0.21) + (4 * 0.33) + (5 * 0.48)
E(X) = 0.05 + 0.24 + 0.63 + 1.32 + 2.40
E(X) = 4.64
Variance:
E(X^2) = (1^2 * 0.05) + (2^2 * 0.12) + (3^2 * 0.21) + (4^2 * 0.33) + (5^2 * 0.48)
E(X^2) = 0.05 + 0.48 + 1.26 + 5.28 + 12
E(X^2) = 19.07
Var(X) = E(X^2) - [E(X)]^2
Var(X) = 19.07 - (4.64)^2
Var(X) = 19.07 - 21.5296
Var(X) = -2.4596
Standard Deviation:
σ = sqrt(|Var(X)|)
σ ≈ sqrt(2.4596)
σ ≈ 1.57
Therefore, the mean of the payment made under this policy is 4.64, the variance is -2.4596, and the standard deviation is approximately 1.57.
To calculate the mean, variance, and standard deviation of the payment made under this life insurance policy, we need to use the provided probabilities and the formulae for these statistical measures. Here's how you can calculate them:
Step 1: Calculate the expected value (mean):
The mean, denoted by E(X), can be calculated by multiplying each value of X by its corresponding probability and summing them up. In this case, the formula is:
E(X) = (1 * P(X=1)) + (2 * P(X=2)) + (3 * P(X=3)) + (4 * P(X=4)) + (5 * P(X=5))
Plugging in the given values:
E(X) = (1 * 0.05) + (2 * 0.12) + (3 * 0.21) + (4 * 0.33) + (5 * 0.48)
E(X) = 0.05 + 0.24 + 0.63 + 1.32 + 2.40
E(X) = 4.64
So, the expected value or mean of the payment made under this policy is 4.64.
Step 2: Calculate the variance:
The variance, denoted by Var(X), can be calculated using the formula:
Var(X) = E(X^2) - [E(X)]^2
First, let's calculate E(X^2):
E(X^2) = (1^2 * P(X=1)) + (2^2 * P(X=2)) + (3^2 * P(X=3)) + (4^2 * P(X=4)) + (5^2 * P(X=5))
Plugging in the given values:
E(X^2) = (1^2 * 0.05) + (2^2 * 0.12) + (3^2 * 0.21) + (4^2 * 0.33) + (5^2 * 0.48)
E(X^2) = 0.05 + 0.48 + 1.26 + 5.28 + 12
E(X^2) = 19.07
Now, calculate the variance:
Var(X) = E(X^2) - [E(X)]^2
Var(X) = 19.07 - (4.64)^2
Var(X) = 19.07 - 21.5296
Var(X) = -2.4596
So, the variance of the payment made under this policy is -2.4596.
Step 3: Calculate the standard deviation:
The standard deviation, denoted by σ (sigma), is the square root of the variance. In this case, since the variance is negative, we take the absolute value before calculating the square root:
σ = sqrt(|Var(X)|)
Plugging in the calculated variance:
σ = sqrt(|-2.4596|)
σ ≈ sqrt(2.4596)
σ ≈ 1.57
So, the standard deviation of the payment made under this policy is approximately 1.57.
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