Probability & Statistics Ch.2 Section 3 Question 11

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Let Y∼Binomial(10,θ)Y \sim \text{Binomial}(10, \theta). Compute P(Y=10)P(Y = 10).

Solution

From Example 2.3.3 in the textbook, we know:

pY(y)=P(Y=y)=(ny)θy(1−θ)n−yp_Y(y) = P(Y=y) = \binom{n}{y}\theta^y(1-\theta)^{n-y}

Where:

  • n = number of trials
  • θ = probability of success
  • y = desired number of successful outcomes

We plug in n = 10, y = 10:

P(Y=10)=(1010)θ10(1−θ)10−10P(Y=10) = \binom{10}{10}\theta^{10}(1-\theta)^{10-10}

P(Y=10)=1⋅θ10⋅(1−θ)0P(Y=10) = 1 \cdot \theta^{10} \cdot (1-\theta)^0

P(Y=10)=θ10P(Y=10) = \theta^{10}