1. The sociologist surveyed the households in a small town. The random variable X represents the number of dependent children in the households. The following is the probability distribution of X:

X 0 1 2 3 4
P(X) 0.07 0.20 0.38 k 0.13
(a) Find the missing probability value of k
(b) Find the mean of X
(c) Find the standard deviation of X

a)k=1-(0.07+0.20+0.38+0.13)=0.22

b)M(X)=0*0.07+1*0.20+2*0.38+3*0.22+4*0.13=
=2.14
c)M(X^2)=0*0.07+1*0.20+4*0.38+9*0.22+
+16*0.13=5.78
The standard deviation=
=sqrt(M(X^2)-M(X)^2)=1.10

(a) To find the missing probability value of k, we can use the fact that the sum of all probabilities must equal 1. So we can set up the following equation:

0.07 + 0.20 + 0.38 + k + 0.13 = 1

Simplifying this equation, we have:

0.78 + k = 1

Subtracting 0.78 from both sides, we find:

k = 0.22

Therefore, the missing probability value of k is 0.22.

(b) The mean of X, denoted by μ, can be calculated using the formula:

μ = ∑(X * P(X))

where X represents each possible value of the random variable and P(X) represents its respective probability.

In this case, we have the following values:

X: 0, 1, 2, 3, 4
P(X): 0.07, 0.20, 0.38, 0.22, 0.13

Calculating the mean using the formula, we get:

μ = (0 * 0.07) + (1 * 0.20) + (2 * 0.38) + (3 * 0.22) + (4 * 0.13)

μ = 1.29

Therefore, the mean of X is 1.29.

(c) The standard deviation of X, denoted by σ, can be calculated using the formula:

σ = sqrt( ∑(x^2 * P(X)) - μ^2 )

where x represents each possible value of the random variable, P(X) represents its respective probability, and μ represents the mean.

For this calculation, we need to find the sum of (x^2 * P(X)) first. Using the same values for X and P(X) as before, we have:

(0^2 * 0.07) + (1^2 * 0.20) + (2^2 * 0.38) + (3^2 * 0.22) + (4^2 * 0.13)

= 0 + 0.20 + 1.52 + 1.98 + 0.52

= 4.22

Now we can substitute this value along with the mean (μ) into the standard deviation formula:

σ = sqrt( 4.22 - 1.29^2 )

= sqrt( 4.22 - 1.6641 )

= sqrt( 2.5559 )

≈ 1.60

Therefore, the standard deviation of X is approximately 1.60.

To find the missing probability value, k, you need to use the fact that the sum of all the probabilities in a probability distribution must equal 1. Therefore, you can set up an equation and solve for k.

Step 1: Write down the sum of all the given probabilities:
0.07 + 0.20 + 0.38 + k + 0.13 = 1

Step 2: Simplify the equation:
0.78 + k = 1

Step 3: Solve for k by subtracting 0.78 from both sides of the equation:
k = 1 - 0.78
k = 0.22

Therefore, the missing probability value, k, is 0.22.

To find the mean of X, you need to multiply each value of X by its corresponding probability and then sum all the results.

Step 1: Multiply each value of X by its probability:
(0 × 0.07) + (1 × 0.20) + (2 × 0.38) + (3 × 0.22) + (4 × 0.13) = 0 + 0.20 + 0.76 + 0.66 + 0.52 = 2.14

Therefore, the mean of X is 2.14.

To find the standard deviation of X, you need to calculate the variance first. The variance is the sum of the squares of the differences between each value of X and the mean, multiplied by their respective probabilities.

Step 1: Calculate the squared differences between each value of X and the mean, and multiply by their probabilities:
[(0 - 2.14)^2 × 0.07] + [(1 - 2.14)^2 × 0.20] + [(2 - 2.14)^2 × 0.38] + [(3 - 2.14)^2 × 0.22] + [(4 - 2.14)^2 × 0.13] = 0.4518 + 0.2144 + 0.0144 + 0.0296 + 0.0548 = 0.765

Step 2: Calculate the standard deviation by taking the square root of the variance:
√0.765 = 0.875

Therefore, the standard deviation of X is approximately 0.875.