The pmf of the amount of memory X (GB) in a purchased flash drive is given as the following.
x	1	2	4	8	16
p(x)	0.05	0.15	0.25	0.45	0.10
(a)
Compute E(X) (in GB). (Enter your answer to two decimal places.)
 GB
(b)
Compute V(X) (in GB2) directly from the definition. (Enter your answer to four decimal places.)
 GB2
(c)
Compute the standard deviation of X (in GB). (Round your answer to three decimal places.)
 GB
(d)
Compute V(X) (in GB2) using the shortcut formula. (Enter your answer to four decimal places.)
 GB2


To solve the problem, we can follow the following steps for each part:

### Part (a) - **Compute \( E(X) \)**

The expected value \( E(X) \) is computed using the formula for the expectation of a discrete random variable:
\[
E(X) = \sum_x x \cdot p(x)
\]
Where \( x \) is the value of the random variable and \( p(x) \) is the probability of each value.

Given values:
\[
\begin{array}{|c|c|}
\hline
x & p(x) \\
\hline
1 & 0.05 \\
2 & 0.15 \\
4 & 0.25 \\
8 & 0.45 \\
16 & 0.10 \\
\hline
\end{array}
\]

Let's compute \( E(X) \):

\[
E(X) = (1 \cdot 0.05) + (2 \cdot 0.15) + (4 \cdot 0.25) + (8 \cdot 0.45) + (16 \cdot 0.10)
\]

### Part (b) - **Compute \( V(X) \)** (Variance)

The variance \( V(X) \) is computed using the formula:
\[
V(X) = E(X^2) - (E(X))^2
\]
Where \( E(X^2) \) is computed as:
\[
E(X^2) = \sum_x x^2 \cdot p(x)
\]
We already know \( E(X) \) from part (a), so we will compute \( E(X^2) \) and then find \( V(X) \).

### Part (c) - **Compute Standard Deviation of \( X \)**

The standard deviation \( \sigma_X \) is the square root of the variance:
\[
\sigma_X = \sqrt{V(X)}
\]

### Part (d) - **Compute \( V(X) \) using the Shortcut Formula**

We will recheck \( V(X) \) using the shortcut formula:
\[
V(X) = E(X^2) - (E(X))^2
\]
Since we will have already calculated both \( E(X^2) \) and \( E(X) \), this is just a verification.

I'll perform the calculations step-by-step.Here are the results of the calculations:

### (a) **Expected Value \( E(X) \)**
\[
E(X) = 6.55 \, \text{GB}
\]

### (b) **Variance \( V(X) \)**
\[
V(X) = 16.1475 \, \text{GB}^2
\]

### (c) **Standard Deviation of \( X \)**
\[
\sigma_X = 4.018 \, \text{GB}
\]

### (d) **Variance \( V(X) \) using the shortcut formula**
This confirms the result from part (b):
\[
V(X) = 16.1475 \, \text{GB}^2
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. How does the probability mass function affect \( E(X) \)?
2. How can the variance help in understanding data dispersion?
3. What does a higher standard deviation indicate about the data spread?
4. How would adding more outcomes to the PMF affect the variance?
5. How would the expectation change if we modify the probabilities?

**Tip**: The variance formula \( V(X) = E(X^2) - (E(X))^2 \) is very useful for simplifying calculations.

The pmf of the amount of memory X (GB) in a purchased flash drive is given as the following.
x    1    2    4    8    16
p(x)    0.05    0.15    0.25    0.45    0.10
(a) Compute E(X) (in GB). (Enter your answer to two decimal places.) GB
(b) Compute V(X) (in GB^2) directly from the definition. (Enter your answer to four decimal places.) GB^2
(c) Compute the standard deviation of X (in GB). (Round your answer to three decimal places.) GB
(d) Compute V(X) (in GB^2) using the shortcut formula. (Enter your answer to four decimal places.) GB^2

Learn how to compute the expected value, variance, and standard deviation for a discrete random variable representing the memory capacity (in GB) of a flash drive. This step-by-step guide includes the probability mass function, key formulas, and detailed solutions. Ideal for undergraduate students.

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