# Sum & Plot Random Variables

The problem you presented requires the determination of the probability density function (pdf) of the sum of two independent random variables: one with a Bernoulli distribution and the other with a standard normal distribution.

Let $X$ be the random variable representing the Bernoulli distribution and $Y$ be the random variable representing the standard normal distribution.

The Bernoulli distribution has the following probability mass function (pmf): $P(X = 0) = 1-p$ $P(X = 1) = p$ where $0 \leq p \leq 1$.

The standard normal distribution has pdf: $f_Y(y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}}$

The distribution of the sum, $Z = X + Y$, will be a convolution of the distributions of $X$ and $Y$. The convolution for continuous and discrete variables can be computed as: $f_Z(z) = \sum_{x} P(X = x) f_Y(z - x)$

Considering the Bernoulli distribution: $f_Z(z) = (1-p) f_Y(z) + p f_Y(z-1)$

To plot this function, we can use a range of values for $z$ and then evaluate $f_Z(z)$ for each value.

Here's Python code that uses `numpy`

and `matplotlib`

to compute and plot the pdf of the sum for a given $p$:

`python````
import numpy as np
import matplotlib.pyplot as plt
# Parameters
p = 0.5 # You can change this value for Bernoulli's p
# Function for the standard normal pdf
def f_Y(y):
return (1/np.sqrt(2*np.pi)) * np.exp(-0.5 * y**2)
# Function for the sum's pdf
def f_Z(z):
return (1-p)*f_Y(z) + p*f_Y(z-1)
# Plot
z_values = np.linspace(-5, 6, 1000) # Range of z values
pdf_values = [f_Z(z) for z in z_values]
plt.plot(z_values, pdf_values)
plt.title("PDF of Sum of Bernoulli and Standard Normal RVs")
plt.xlabel("z")
plt.ylabel("Probability Density")
plt.grid(True)
plt.show()
```

This code plots the pdf of the sum of a Bernoulli and a standard normal random variable for $p = 0.5$. You can change the value of $p$ in the code to see the effect on the resulting pdf.

I'm sorry, but I cannot directly produce or output images. However, I've provided the Python code above which you can run locally on your machine. If you run the code using Python, `numpy`

, and `matplotlib`

, you'll be able to visualize the plot.

To save the plot as a PNG or JPEG:

- Add this before
`plt.show()`

to save it as a PNG:

`python````
plt.savefig("result_plot.png")
```

- Or, for a JPEG:

`python````
plt.savefig("result_plot.jpg", format='jpeg')
```

Once you run the script, it will generate and save the plot in the same directory as the script.