government data assign a single case or each death that occurs in the u.s. the data show that the probability is 0.45 that a randomly chosen death was due to cardiovascular disease, and 0.22 that i was due to cancer. Suppose a docote checked the cause of death of 2 randomly selected patients. Define a random variable X as the number of patients whose cause of death was cardiovascular disease

What is the sample space of this random process?

What is the probability for the event {DD}, both were dead from cardiovascular disease.

What is P(x=0)?
What is the probability distribution of x?
mean for x?
standard dievation or x?

To understand the random process and answer the questions, let's break it down step by step.

1. Sample Space:
The sample space refers to all possible outcomes of a random process. In this case, the doctor checked the cause of death for two randomly selected patients. Since each patient can either die from cardiovascular disease (C) or not (N), the sample space for this random process consists of the following outcomes:
{CC, CN, NC, NN}

2. Probability of both deaths due to cardiovascular disease:
To find the probability that both patients died from cardiovascular disease, we need to look at the given information. The probability of a randomly chosen death being due to cardiovascular disease is 0.45. Since we are dealing with two patients, we can multiply the probabilities together:
P({DD}) = P(D_1 = C) * P(D_2 = C) = 0.45 * 0.45 = 0.2025

3. P(x=0):
Here, x represents the number of patients whose cause of death was cardiovascular disease. So, when x=0, it means neither patient died from cardiovascular disease. Looking at the sample space, we see two outcomes where x=0: {NN, NC}. To find P(x=0), we add the probabilities of these two outcomes:
P(x=0) = P(NN) + P(NC) = 0.55 * 0.55 + 0.55 * 0.45 = 0.3025 + 0.2475 = 0.55

4. Probability distribution of x:
To find the probability distribution of x, we need to consider all possible values x can take, along with their respective probabilities. In this case, x can only take values 0 or 1 since there are only two patients. The probability distribution for x is as follows:
P(x=0) = 0.55
P(x=1) = 0.45

5. Mean for x:
The mean of a probability distribution represents the average value. To find the mean for x, we multiply each value of x by its respective probability and sum them up:
Mean (μ) = x * P(x)
Mean (μ) = 0 * 0.55 + 1 * 0.45 = 0 + 0.45 = 0.45

6. Standard deviation for x:
The standard deviation measures the dispersion or spread of a probability distribution. To calculate the standard deviation for x, we need to find the variance first. The variance (σ^2) is given by the following formula:
Variance (σ^2) = (x - μ)^2 * P(x)
Variance (σ^2) = (0 - 0.45)^2 * 0.55 + (1 - 0.45)^2 * 0.45 = 0.2025 * 0.55 + 0.3025 * 0.45 = 0.111375 + 0.136125 = 0.2475
Finally, the standard deviation (σ) is the square root of the variance:
Standard Deviation (σ) = √(0.2475) ≈ 0.4975

These calculations provide the necessary information about the random variable X and the associated probabilities for the given scenario.