Find the variance for the number of boys in a royal family that has children until there is a boy or until there are three children, whichever comes first. Assume the probability of having a boy is 1/2.

• 0.06
• 0.08
• 0.11
• 0.14
• 0.17

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Let [math]X[/math] be a random variable with [math]E(X) = \mu[/math] and [math]Var(X) = \sigma^2[/math]. Suppose

for all [math]c[/math]. Determine [math]E[(X-c_0)^2] [/math].

• [math]\frac{\sigma^2}{2}[/math]
• [math]\mu^2[/math]
• [math]\sigma^2[/math]
• [math]\min(\sigma^2,\mu^2)[/math]
• [math]\frac{\sigma^2 + \mu^2}{2}[/math]

For a constant [math]b = 400[/math], you are given the following about a loss [math]L[/math]:

• [math]\operatorname{E}[L^2 \cdot 1_{L \leq b} ] = 3[/math]
• [math]\operatorname{E}[L \cdot 1_{L \leq b} ] = 2[/math]
• [math]\operatorname{P}(L \leq b) =0.8[/math]

Determine the variance of the payment when a limit of [math]b[/math] is applied.

• 25,276
• 25,279
• 25,603
• 32,003
• 32,103

Ten cards from a deck of playing cards are in a box: two diamonds, three spades, and five hearts. Two cards are randomly selected without replacement.

Calculate the variance of the number of diamonds selected, given that no spade is selected.

• 0.24
• 0.28
• 0.32
• 0.34
• 0.41

The distribution of values of the retirement package offered by a company to new employees is modeled by the probability density function

Calculate the variance of the retirement package value for a new employee, given that the value is at least 10.

• 15
• 20
• 25
• 30
• 35

Let [math]X[/math] and [math]Y[/math] be two random variables defined on the finite sample space [math]\Omega[/math]. Assume that [math]X[/math], [math]Y[/math], [math]X + Y[/math], and [math]X - Y[/math] all have the same distribution. Determine [math]P(X = Y = 0) [/math].

• 0
• 0.2
• 0.5
• 0.8
• 1

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.