Probability distribution of winning ticket

[hushed garden]

Started by trying to find pmf that winning ticket number = x by summing each possible number of tickets sold and multiplying each by uniform distribution, but idk where to go now

[novel tundra]

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[hushed garden]

thanks in advance to anyone who can help!

[feral ether]

for part (a) you have to use the formula

,,E[X] = E[E[X|N]]

[ornate anchor]

vin100

[feral ether]

so u shld b finding $P(X = x \mid N = n)$ first

[ornate anchor]

vin100

[feral ether]

,,X \sim \mathrm{Unif}({1, \dots, N})

[ornate anchor]

vin100

[feral ether]

so $P(X = x \mid N = n) = \frac1n$

[ornate anchor]

vin100

[feral ether]

for all $x \in {1, \dots, n}$

[ornate anchor]

vin100

[feral ether]

,,E[X \mid N = n] = \sum_{x = 1}^n x \cdot \frac1n = \frac{n+1}{2}

[ornate anchor]

vin100