Given Event A and Event B , calculate $P(A \cup B)$ if $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$.

Using the Addition Law : $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.5 - 0.1 = 0.8$.

Determine the conditional probability $P(A|B)$ if $P(A \cap B) = 0.15$ and $P(B) = 0.30$.

The formula for conditional probability is $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Therefore: $P(A|B) = \frac{0.15}{0.30} = 0.5$.

If a process follows a Poisson distribution with an average rate of $\lambda = 3$ arrivals per hour, what is the variance $\sigma^2$?

For a Poisson distribution, the mean and variance are equal. Therefore, $\sigma^2 = \lambda = 3$.

Calculate the Z-score for a value $x = 110$ from a population with mean $\mu = 100$ and standard deviation $\sigma = 5$.

The Z-score formula is $Z = \frac{x - \mu}{\sigma}$. Substituting the values: $Z = \frac{110 - 100}{5} = \frac{10}{5} = 2$.

Identify whether a random variable representing the 'number of customer complaints per day' is discrete or continuous .

The variable is discrete because the number of complaints can only take on countable, integer values (0, 1, 2, ...).

If events $X$ and $Y$ are mutually exclusive , what is the value of $P(X \cap Y)$?

For mutually exclusive events, they cannot occur simultaneously, therefore $P(X \cap Y) = 0$.

Given $n=5$ and $p=0.5$, calculate the probability $P(X=0)$ using the Binomial distribution formula $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$.

$P(X=0) = \binom{5}{0} (0.5)^0 (0.5)^5 = 1 \cdot 1 \cdot 0.03125 = 0.03125$.

Which Excel function should be used to find the probability of a value being less than or equal to $x$ in a Normal distribution ?

Use the NORM.DIST function: `NORM.DIST(x, mean, standard_dev, TRUE)`.

If a Poisson rate is $\lambda = 2$ per hour, what is the adjusted rate $\lambda'$ for a 30-minute interval ?

Since 30 minutes is 0.5 hours, the adjusted rate is $\lambda' = 2 \cdot 0.5 = 1$.

What is the total area under the curve of a Standard Normal distribution ?

The total area under the probability density function for any Normal distribution (including the standard one) is exactly 1.

Determine the variance $\sigma^2$ of a Binomial distribution where $n=20$ and $p=0.4$.

The variance is given by $\sigma^2 = n \cdot p \cdot (1-p)$. Thus: $\sigma^2 = 20 \cdot 0.4 \cdot 0.6 = 4.8$.

Explain the significance of the Standard Normal distribution regarding the Z-score .

The Standard Normal distribution has a mean $\mu = 0$ and standard deviation $\sigma = 1$. Standardizing any normal variable to a Z-score allows the use of a single Z-table or standardized function to compute probabilities for any normal distribution.

If $P(A|B) = P(A)$, what can you conclude about events A and B ?

The events $A$ and $B$ are statistically independent , meaning the occurrence of $B$ provides no information about the likelihood of $A$.

Which Excel function is used to find the value $x$ corresponding to a specific percentile in a Normal distribution ?

Use the NORM.INV function: `NORM.INV(probability, mean, standard_dev)`.

Probability Theory And Statistical Distributions For Financial And Business Analytics Flashcards

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Flashcard Sets

Probability Theory and Statistical Distributions for Financial and Business Analytics

This deck covers essential probability theory, discrete and continuous distributions, and their practical application in modeling business uncertainty and financial risks.

15 cards

intermediate

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by @soumavo_bhattacharya

Q: Given Event A and Event B, calculate $P(A \cup B)$ if $P(A) = 0.4$...

Q: Determine the conditional probability $P(A|B)$ if $P(A \cap B) = 0.15$ and $P(B) = ...

Q: In a Binomial distribution experiment with $n=10$ trials and success probability $p...

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