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 $p$:

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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:

1. Add this before plt.show() to save it as a PNG:

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plt.savefig("result_plot.png")

2. Or, for a JPEG:

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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.