Recall that if $X$ is a Bernoulli random variable with parameter $p$, then the expectation $E[X]$ and the variance $V(X)$ of $X$ are given by
\begin{align*}
E[X] &= p \\
V(X) &= p(1-p).
\end{align*}
Recall (see the hint above) that the variance of a Bernoulli random variable $X$ with parameter $p$ is
\[V(X) = p(1-p).\]
Thus we have
\[p(1-p) = 0.21.\]
This yields the quadratic equation
\[p^2-p+0.21 = (p-0.3)(p-0.7) = 0.\]
Solving the equation, we obtain $p = 0.3$ or $p = 0.7$. By assumption, $p > 0.5$. Hence we conclude that $p = 0.7$.
Solution of (b)
The expectation $E(X)$ of a Bernoulli distributed $X$ is given by $E(X) = p$ (see the hint above). As we obtained $p=0.7$ in Part (a), we see that the expectation is $E(X) = 0.7$.
Expectation, Variance, and Standard Deviation of Bernoulli Random Variables
A random variable $X$ is said to be a Bernoulli random variable if its probability mass function is given by
\begin{align*}
P(X=0) &= 1-p\\
P(X=1) & = p
\end{align*}
for some real number $0 \leq p \leq 1$.
(1) Find the expectation of the Bernoulli random variable $X$ […]
How to Prove Markov’s Inequality and Chebyshev’s Inequality
(a) Let $X$ be a random variable that takes only non-negative values. Prove that for any $a > 0$,
\[P(X \geq a) \leq \frac{E[X]}{a}.\]
This inequality is called Markov's inequality.
(b) Let $X$ be a random variable with finite mean $\mu$ and variance $\sigma^2$. Prove that […]
Upper Bound of the Variance When a Random Variable is Bounded
Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
\[V(X) \leq \frac{c^2}{4}.\]
Proof.
Recall that […]
Can a Student Pass By Randomly Answering Multiple Choice Questions?
A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and […]
Expected Value and Variance of Exponential Random Variable
Let $X$ be an exponential random variable with parameter $\lambda$.
(a) For any positive integer $n$, prove that
\[E[X^n] = \frac{n}{\lambda} E[X^{n-1}].\]
(b) Find the expected value of $X$.
(c) Find the variance of $X$.
(d) Find the standard deviation of […]
Linearity of Expectations E(X+Y) = E(X) + E(Y)
Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as
\[E(X+Y) = E(X) + E(Y).\]
Solution.
The joint probability mass function of the discrete random variables $X$ and $Y$ is defined by
\[p(x, y) = P(X=x, Y=y).\]
Note that the […]
Coupon Collecting Problem: Find the Expectation of Boxes to Collect All Toys
A box of some snacks includes one of five toys. The chances of getting any of the toys are equally likely and independent of the previous results.
(a) Suppose that you buy the box until you complete all the five toys. Find the expected number of boxes that you need to buy.
(b) […]
How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions
Let $X\sim \mathcal{N}(\mu, \sigma)$ be a normal random variable with parameter $\mu=6$ and $\sigma^2=4$. Find the following probabilities using the Z-table below.
(a) Find $P(X \lt 7)$.
(b) Find $P(X \lt 3)$.
(c) Find $P(4.5 \lt X \lt 8.5)$.
The Z-table is […]