Discrete Distributions - GRE Subject Test: Math

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Question

A fair coin is tossed 15 times. What is the probability of observing less than 3 heads?

Answer

This problem uses the Binomial Distribution: $\binom{n}{k}p^k(1-p)^{n-k}$

For this problem n is the number of trials, or 15. Because the problem stated that the coin was a fair coin the probability of heads is one half, or .5.

The binomial distribution is a discrete distribution so the expression x<3 has to be broken down.

$Pr(x=0)=\begin{pmatrix} 15\0 \end{pmatrix} (.5)^0(1-.5)^{15}=.000031$

$Pr(x=1)=\begin{pmatrix} 15\1 \end{pmatrix} (.5)^1(1-.5)^{14}=.00046$

$Pr(x=2)=\begin{pmatrix} 15\2 \end{pmatrix} (.5)^2(1-.5)^{13}=.0032$

Adding the probabilities will give the final answer.

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Question

X is a continuously and uniformly distributed on the interval (0,50). Find the Expected Value (E\[x\]) and Variance (Var(x)) of X.

Answer

Because x is a continuous uniform random variable the expected value and variance can be found with the following formulas:

$E[x]=\frac{a+b}{2}$

$Var(x)=\frac{(b-a)^2}{12}$

X is uniform on (a,b). In this case a is 0 and b is 50. Plugging the values of a and b into the given formulas will give the answers:

$E[x]=\frac{a+b}{2}=\frac{0+50}{2}=25$

$Var(x)=\frac{(b-a)^2}{12}=\frac{2500}{12}=208.33$

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