Five males with a particular genetic disorder have one child each. The random variable x is the number of children among the five who inherit the genetic disorder. Determine whether the table describes a probability distribution. If it does, find the mean and standard deviation.
x
0
1
2
3
4
5
P(x)
0.0010
0.0146
0.0879
0.2637
0.3955
0.2373
Find the mean of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
muequals
(Round to one decimal place as needed.)
B.
The table is not a probability distribution.
Find the standard deviation of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
sigmaequals
(Round to one decimal place as needed.)
B.
The table is not a probability distribution.
ok Can you please help me?
To determine whether the table describes a probability distribution, we need to check if the probabilities sum up to 1. Let's calculate the sum of the probabilities:
0.0010 + 0.0146 + 0.0879 + 0.2637 + 0.3955 + 0.2373 = 1
Since the probabilities sum up to 1, the table is a probability distribution.
To find the mean (average) of the random variable x, we multiply each value of x by its corresponding probability, and then sum up these products.
Mean (μ) = (0 * 0.0010) + (1 * 0.0146) + (2 * 0.0879) + (3 * 0.2637) + (4 * 0.3955) + (5 * 0.2373)
= 0 + 0.0146 + 0.1758 + 0.7911 + 1.582 + 1.1865
= 3.75 (rounded to two decimal places)
Therefore, the mean of the random variable x is 3.75.
To find the standard deviation of the random variable x, we need to calculate the variance first. The variance (σ^2) is calculated by finding the squared difference of each value of x from the mean, multiplying it by its corresponding probability, and summing up these products.
Variance (σ^2) = [(0 - 3.75)^2 * 0.0010] + [(1 - 3.75)^2 * 0.0146] + [(2 - 3.75)^2 * 0.0879] + [(3 - 3.75)^2 * 0.2637] + [(4 - 3.75)^2 * 0.3955] + [(5 - 3.75)^2 * 0.2373]
= [(-3.75)^2 * 0.0010] + [(-2.75)^2 * 0.0146] + [(-1.75)^2 * 0.0879] + [(-0.75)^2 * 0.2637] + [(-0.25)^2 * 0.3955] + [(1.25)^2 * 0.2373]
= 14.0625 * 0.0010 + 7.5625 * 0.0146 + 3.0625 * 0.0879 + 0.5625 * 0.2637 + 0.0625 * 0.3955 + 1.5625 * 0.2373
= 0.0140625 + 0.1100575 + 0.26746875 + 0.14786375 + 0.02471875 + 0.37096875
= 0.93514 (rounded to five decimal places)
The standard deviation (σ) is the square root of the variance.
Standard deviation (σ) = √(0.93514)
= 0.9663 (rounded to four decimal places)
Therefore, the standard deviation of the random variable x is 0.9663.
To determine whether the table describes a probability distribution, we need to check if the probabilities sum up to 1 and if all probabilities are non-negative.
Let's calculate the sum of the probabilities:
0.0010 + 0.0146 + 0.0879 + 0.2637 + 0.3955 + 0.2373 = 1
The sum of the probabilities is 1, which means the table describes a probability distribution.
Now, let's calculate the mean of the random variable x. The mean is calculated by multiplying each value of x by its corresponding probability and summing them up.
Mean (μ) = Σ (x * P(x))
0 * 0.0010 + 1 * 0.0146 + 2 * 0.0879 + 3 * 0.2637 + 4 * 0.3955 + 5 * 0.2373 = 2.5
So, the mean of the random variable x is 2.5.
Now let's calculate the standard deviation of the random variable x. The standard deviation is a measure of the dispersion of the values around the mean and is calculated using the formula:
Standard Deviation (σ) = sqrt(Σ [(x - μ)^2 * P(x)])
0.0010 * (0 - 2.5)^2 + 0.0146 * (1 - 2.5)^2 + 0.0879 * (2 - 2.5)^2 + 0.2637 * (3 - 2.5)^2 + 0.3955 * (4 - 2.5)^2 + 0.2373 * (5 - 2.5)^2
Calculating this expression gives us approximately 1.23.
So, the standard deviation of the random variable x is approximately 1.23.
Therefore, the correct choices are:
A. The mean (µ) equals 2.5
A. The standard deviation (σ) equals 1.23.