# 1. Introduction

This R program calculates the mean, variance, and standard deviation for a given discrete probability distribution. A discrete probability distribution lists the probabilities of occurrence of different outcomes of a discrete random variable.

# 2. Program Steps

1. Define the outcomes of the discrete random variable.

2. Define the probabilities associated with each outcome.

3. Calculate the mean of the distribution.

4. Calculate the variance of the distribution.

5. Calculate the standard deviation of the distribution.

6. Print the mean, variance, and standard deviation.

# 3. Code Program

``````# Step 1: Define the outcomes of the discrete random variable
outcomes <- c(1, 2, 3, 4, 5)

# Step 2: Define the probabilities of each outcome
probabilities <- c(0.1, 0.2, 0.3, 0.2, 0.2)

# Step 3: Calculate the mean
mean_value <- sum(outcomes * probabilities)

# Step 4: Calculate the variance
variance_value <- sum((outcomes - mean_value)^2 * probabilities)

# Step 5: Calculate the standard deviation
sd_value <- sqrt(variance_value)

# Step 6: Print the results
print("Mean of the distribution:")
print(mean_value)
print("Variance of the distribution:")
print(variance_value)
print("Standard deviation of the distribution:")
print(sd_value)

``````

### Output:

```Mean of the distribution:
[Calculated Mean]
Variance of the distribution:
[Calculated Variance]
Standard deviation of the distribution:
[Calculated Standard Deviation]
```

### Explanation:

1. outcomes <- c(1, 2, 3, 4, 5): Specifies the outcomes of the discrete random variable.

2. probabilities <- c(0.1, 0.2, 0.3, 0.2, 0.2): Defines the probability associated with each outcome.

3. sum(outcomes * probabilities): Calculates the mean of the distribution.

4. sum((outcomes - mean_value)^2 * probabilities): Calculates the variance of the distribution.

5. sqrt(variance_value): Calculates the standard deviation, which is the square root of the variance.

6. print("Mean of the distribution:"), print(mean_value): Prints the mean of the distribution.

7. print("Variance of the distribution:"), print(variance_value): Prints the variance of the distribution.

8. print("Standard deviation of the distribution:"), print(sd_value): Prints the standard deviation of the distribution.