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 $X$ […]

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(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 […]

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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 […]

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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 […]

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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 […]

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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.
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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 […]